---
title: A Moving Target for Erdos Problem 125
lead: The sums of a base-3 number and a base-4 number, each written with the digits `0` and `1` only, have upper density at least `6/(11 liminf_k h_k(tau_k))`, where `h_k` is the cell energy of a Cantor convolution read at the scaling `tau_k = 4^(m_k)/3^k`; so the open half of Erdos problem 125 fails only if one orbit of an irrational rotation lands, at every late step, in a target of measure at most `190/M`, for every `M`.
date: 2026-09-23
figure: paper-erdos-125-upper-density
---

Add a number written in base `3` with the digits `0` and `1` only to a number written in base `4` with the same two digits. Erdos problem 125 asks whether the sums fill a positive proportion of the integers. In the lower sense they do not: an argument found by DeepMind and checked in Lean shows that the proportion below `x` falls under any `eps > 0` for arbitrarily large `x`. Whether it rises above one fixed `c > 0` for arbitrarily large `x`, positive upper density, is open, and it is the variant the formal statement of the problem leaves as `answer(sorry)`. This paper does not settle it; it changes its shape. An exact identity carries the sums below `3^k + 4^m` onto a continuous object, the sumset of two Cantor sets at the scaling `tau = 4^m/3^k`, read on cells of side `3^(-k)`. Across that bridge the energy ratio that bounds the count from below is subpolynomial, by Shmerkin's theorem on `L^q` dimensions, and unbounded along infinitely many pairs, by the resonances of Nazarov, Peres and Shmerkin; its average over the scaling is at most `190`; and the upper density is at least `6/(11 liminf_k h_k(tau_k))`, where `h_k` is the normalized cell energy and the scalings `tau_k` walk the circle `[1, 4)` by an irrational rotation. So the upper density is positive unless that one orbit lands, at every late step, in a target of measure at most `190/M`, for every `M`. Almost every rotated orbit misses such targets infinitely often, and the census of the energy ratio along the orbit to `k = 29` keeps it between `1.63` and `2.01`; the orbit that matters is a single one, and neither fact reaches it.

## Introduction

Let `A` be the integers whose base-3 digits are all `0` or `1`, `0, 1, 3, 4, 9, 10, 12, 13, 27, ...`, and `B` those whose base-4 digits are, `0, 1, 4, 5, 16, 17, 20, 21, 64, ...`. Each is thin: `A` has about `x^0.631` members below `x` and `B` about `x^(1/2)`. Their sumset `S = A + B` is not. Every integer up to `61` lies in it, the first two it misses are `62` and `63`, and below `3^22` it holds `26666749554` integers, `85` percent of them (Fact 11.1). Erdos problem 125 asks whether `S` has positive lower density; Burr, Erdos, Graham and Li had asked for positive density and for positive upper density. The lower density is `0`, and the formal statement of the problem keeps the other half open:

```
theorem erdos_125.variants.positive_upper_density :
    answer(sorry) <-> 0 < (A + B).upperDensity
```

This paper is about that statement: with `D(x) = card(S meet [1, x])/x`, is `limsup_(x -> infinity) D(x) > 0`?

The first gap is typical. A sum `a + b` with `a < 81` and `b < 64` is at most `40 + 21 = 61`, and a sum with `a >= 81` or `b >= 64` is at least `64`, so nothing lands on `62` or `63`. The same happens at every pair of levels: `S` misses every integer strictly between `d(k, m) = (3^k - 1)/2 + (4^m - 1)/3`, the largest sum of the lower levels, and `min(3^k, 4^m)`. Where a power of `3` and a power of `4` nearly coincide, that gap swallows about a sixth of the range and the density dips: its lowest reading below `3^22`, `0.763391`, sits just below `3^15`, where `4^12/3^15 = 1.169234`. Compounding the dip along ever closer coincidences drives the lower density to `0`, and that is the known answer. For the upper density the question is whether, between the coincidences, the density recovers to one fixed positive level infinitely often.

Counting a sumset from below goes through collisions. If the `2^(k+m)` pairs `(a, b)` of the two lower levels rarely share a sum, their sums are many: Cauchy-Schwarz gives `card(A_k + B_m) >= (d + 1)/Q(k, m)`, where the energy ratio `Q` counts the quadruples with `a + b = a' + b'` against the count a flat distribution on `[0, d]` would give. So a bound on `Q` along infinitely many pairs gives positive upper density. The first observation here is that `Q` is exactly a continuum quantity: `A_k + B_m` is the sumset `C_3 + tau C_4` of two Cantor sets at the one scaling `tau = 4^m/3^k`, read on cells of side `3^(-k)`, and `Q` is the energy of the natural measure on it at that resolution (Theorem 3.1). The figure draws that energy.

![Seven curves over the circle of scalings tau from 1 to 4 on a logarithmic axis, the normalized cell energy h_k(tau) of the base-3 Cantor measure plus tau times the base-4 Cantor measure at resolutions k = 0 to 6, nested from the dim flat-topped curve of k = 0 at the bottom to the jagged blue curve of k = 6 at the top, all falling from left to right; seven orange dots, one on each curve, at the chain points tau_k = 4^(m_k)/3^k, 1, 4/3, 16/9, 64/27, 256/81, 256/243 and 1024/729, each with a hairline drop to the axis.](paper-erdos-125-upper-density)

The horizontal axis is the circle of scalings `tau in [1, 4)`, drawn on a logarithmic scale so that multiplying by `4` goes once around. Each curve is the normalized cell energy `h_k(tau) = 3^k sum_x rho_tau(c_x)^2` of Definition 3.1 at one resolution, `k = 0` dim at the bottom to `k = 6` blue at the top; a flat measure on the support would read `1/(1/2 + tau/3)`, and a larger value means more collisions and fewer sums. The curves nest, because refining a cell never lowers the energy (move (iv) of Theorem 3.4), and their peaks sit near ratios `4^m/3^j` of small exponents, where the two bases coincide early. The orange dots are the chain. At level `k` the pair that decides the density is `(k, m_k)`, with `m_k` the least exponent such that `4^(m_k) >= 3^k`; its scaling `tau_k = 4^(m_k)/3^k` lies in `[1, 4)`, and the dot sits at `(tau_k, h_k(tau_k))` on the curve of its own level. From one level to the next `tau_k` turns by `-log_4 3` around the circle, an irrational rotation, so the dots walk the circle without repeating. At every dot the value is exact, `h_k(tau_k) = 3^k E(k, m_k)/4^(k + m_k)` with `E` the integer energy of `mrlyrs::num::sumset::Pair::energy`, and the binary that draws the figure asserts it.

Theorem 3.6 makes the dots decisive: the upper density of `S` is at least `6/(11 liminf_k h_k(tau_k))`, so it is positive unless the dots climb to infinity. Theorem 3.5 makes the high ground small: `h_k` integrates to at most `190` over `[1, 4]` at every level, so the scalings with `h_k > M` fill a set of measure at most `190/M`. Zero upper density would therefore force one orbit of an irrational rotation into shrinking targets at every late step. Almost every rotated copy of the orbit escapes infinitely often (Theorem 3.7), and along the orbit itself the census keeps `Q(k, m_k)` inside `[1.638124, 2.004783]` for every `6 <= k <= 29` (Fact 11.2); but the orbit that matters is one orbit, and no statement about almost every orbit reaches it. The question stays open.

## What is in print

Throughout, `A_k = A meet [0, 3^k)` and `B_m = B meet [0, 4^m)` are the two levels, of `2^k` and `2^m` integers, and `d(k, m) = (3^k - 1)/2 + (4^m - 1)/3` is the largest element of `A_k + B_m`. Every source below is listed in the references. The problem page and its comment thread, the formal statement, Shmerkin (2019), Nazarov, Peres and Shmerkin (2012) and Glasscock, Moreira and Richter (2024) are read at source; Hasler and Melfi (2024) is read in its abstract and through the problem page; Burr, Erdos, Graham and Li (1996) is not read at source.

**The problem.** The problem page states the question for positive lower density, attributes it to Burr, Erdos, Graham and Li, who ask "positive density? positive upper density?", and marks it disproved, with the proof verified in Lean. It credits an argument found by DeepMind that first shows the upper density to be at least `6/5` times the lower density, so that `S` has no positive density, and then the lower density to be `0`: for every `eps > 0` there are infinitely many `x` with `card(S meet [1, x]) < eps x`. The formal statement in the formal-conjectures repository lists the four possible patterns of the two densities, records the lower-density question as solved and keeps `erdos_125.variants.positive_upper_density` open with `answer(sorry)`; the comment thread on the problem page calls it the likely harder half.

**The gap and the injection.** The argument, as the comment thread reads the Lean proof, rests on one elementary fact.

**Lemma 2.1 (the gap).** For all `k, m >= 0`, `S` misses the open interval `(d(k, m), min(3^k, 4^m))`.

**Proof.** A sum `a + b` with `a < 3^k` and `b < 4^m` has `a in A_k` and `b in B_m`, so it is at most `d(k, m)`. A sum with `a >= 3^k` or `b >= 4^m` is at least `min(3^k, 4^m)`. □

Take a scale `L` with `3^k` and `4^m` both in `[L, (1 + eps) L]`. Then `S meet [0, L)` lies in `[0, d(k, m)]`, inside `[0, (5/6)(1 + eps) L]`. Split each `x < L N` of `S` along one chosen representation `x = a + b`, as `a = 3^k a_1 + a_0` and `b = 4^m b_1 + b_0` with `a_0 in A_k`, `b_0 in B_m`, `a_1 in A` and `b_1 in B`, and put `y = a_1 + b_1`, a member of `S` below `N`. Then `x - L y = a_1 (3^k - L) + b_1 (4^m - L) + a_0 + b_0` lies in `[0, eps y L + (5/6)(1 + eps) L]`, inside `[0, (5/6 + delta) L)` once `eps <= delta/(N + 5/6)`, and `x -> (y, x - L y)` is injective. So `card(S meet [0, L N)) <= card(S meet [0, N)) ((5/6 + delta) L + 1)`, and `D(L N)` is at most about `(5/6 + delta) D(N)` once `L` is large. Since `log 4/log 3` is irrational, such scales exist for every `eps`, and iterating along them drives `D` to `0` on a sequence. The iteration needs very close coincidences, `abs(m log 4 - k log 3) < delta/N`, and no `k <= 22` supplies one close enough to show in the census of Section 11, which sees the single gaps and nothing compounding.

**Growth.** Hasler and Melfi (2024) prove `card(S meet [1, x]) >> x^0.97777`, improving an older estimate of Melfi; the problem page also quotes from them the bound `1015/1458 = 0.69616` from above on the lower density, now superseded by `0`. Glasscock, Moreira and Richter (2024), Theorem C, prove from Shmerkin's uniform `L^q` bounds that `A' + B'` has mass dimension `min(1, dim A' + dim B')` for every `x3`-invariant set of integers `A'` and every `x4`-invariant `B'`, `dim` being the limit of `log card(A' meet [0, N))/log N`. At this pair that is `card(S meet [1, x]) = x^(1 - o(1))`, past `x^0.97777`. Their question on positive density for sumsets of full dimension asks for positive upper density in that generality, and `S` is a case of it.

**The continuum.** Write `mu` and `nu` for the laws of `sum_(l >= 1) X_l 3^(-l)` and `sum_(l >= 1) Y_l 4^(-l)`, the digits `X_l`, `Y_l` independent and uniform on `{0, 1}`, and `C_3` in `[0, 1/2]` and `C_4` in `[0, 1/3]` for their supports, Cantor sets of dimension `log_3 2 = 0.630930` and `1/2`. Shmerkin (2019), Theorem 1.11, proves that for a pleasant model with exponential separation, whose finitely supported driving measures depend continuously on the point outside a null set and have a bounded number of atoms, the `L^q` dimension of every measure of the model exists, the limit uniform over the model, and equals an explicit `min(D_q, 1)`. His Lemma 7.1 and the proof of his Theorem 7.2 make the convolutions of two homogeneous self-similar measures, with an irrational ratio of the logarithms of the contractions and a separation hypothesis on each, such a model over a circle that covers one full period of the scaling, the driving measure there having at most four atoms and one discontinuity; at this pair `D_2 = log_3 2 + 1/2 > 1`. Nazarov, Peres and Shmerkin (2012), Theorem 1.1, had proved the correlation dimension `min(d_a + d_b, 1)`, `d_a = log 2/log(1/a)`, for the natural measures of the central Cantor sets of ratios `a` and `b` convolved at every nonzero scaling when `log b/log a` is irrational. Their Theorem 4.1 makes such a convolution singular on a dense `G_delta` set of scalings whenever `1/a` and `1/b` are Pisot numbers, names `a = 1/4`, `b = 1/3`, this pair, as the example, and its proof finds the Fourier transform away from `0` at each resonance `abs(lambda 4^n - 3^m) < 1/4` of their scaling `lambda`, in their symmetric coordinates. The projection theorem of Marstrand (1954), applied to the product `C_3 x C_4` of dimension `log_3 2 + 1/2 > 1`, gives `leb(C_3 + tau C_4) > 0` for almost every `tau`. Glasscock, Moreira and Richter state a question of Hochman on the Lebesgue measure of `X + Y` for `xr`- and `xs`-invariant closed sets `X`, `Y` whose dimensions add past `1`; it is read here in their paraphrase and not in the original. None of Shmerkin, Nazarov, Peres and Shmerkin, or Glasscock, Moreira and Richter is cited on the problem page or in its thread as read here.

**The complement.** OEIS A367090 lists the integers outside `S`, `62, 63, 143, 144, ...`, and records the reflection `x -> d - x` of `S meet [0, d]` as a proposition on the window `1 < 4^m/3^k <= 4/3`; Lemma 4.1 below carries it to every clean pair.

## Definitions and results

**Definition 3.1.** Fix levels `k, m >= 0` and write `d = d(k, m)`.

- `r(x) = card{(a, b) in A_k x B_m : a + b = x}`, `E(k, m) = sum_x r(x)^2` the additive energy, and `Q(k, m) = E(k, m) (d + 1)/4^(k+m)` the energy ratio, the energy against its value `4^(k+m)/(d + 1)` for a flat `r` on `[0, d]`.
- The pair `(k, m)` is clean when `3^k > d` and `4^m > d`, and a gap copy when `2 4^m < 3^k + 5` or `3^(k+1) < 4^m + 5`.
- `rho_tau` is the law of `a + tau b`, `a` from `mu` and `b` from `nu` independent, for a scaling `tau > 0`; its support is `C_3 + tau C_4`, inside `[0, 1/2 + tau/3]`.
- The cells of level `k` are `c_x = [x 3^(-k), (x + 1) 3^(-k))`, `x` an integer, and `h_k(tau) = 3^k sum_x rho_tau(c_x)^2` is the cell energy; `G_k(M)` is the set of `tau in [1, 4]` with `h_k(tau) > M`.
- The chain: `m_k` is the least integer with `4^(m_k) >= 3^k`, `tau_k = 4^(m_k)/3^k`, which lies in `[1, 4)`, and `d_k = d(k, m_k)`.
- `D(x) = card(S meet [1, x])/x`; the upper density of `S` is `limsup_(x -> infinity) D(x)`.

**Theorem 3.1 (the bridge).** At `tau = 4^m/3^k`, `rho_tau(c_x) = r(x)/2^(k+m)` for every integer `x`. Hence `Q(k, m) = (d + 1) sum_x rho_tau(c_x)^2 = (d + 1) 3^(-k) h_k(tau)`, `card(A_k + B_m)` is exactly the number of cells `c_x` that meet `C_3 + tau C_4`, and `leb(C_3 + tau C_4) <= (5/6) 3^(-k) card(A_k + B_m)`. Because the lower density of `S` is `0`, the infimum of `leb(C_3 + tau C_4)` over `tau in [1, 4)` is `0`, approached along the chain.

**Theorem 3.2 (the energy ratio is subpolynomial).** For every `eps > 0` there is `k_0` with `Q(k, m) <= 3^(eps k)` for all `k >= k_0` and all `m` with `1/3 <= 4^m/3^k < 4`, a window holding every clean pair. Hence `card(S meet [1, x]) >= x^(1 - eps)` for all large `x`.

**Theorem 3.3 (the energy ratio is unbounded).** There is an infinite family of clean pairs along which `Q(k, m) -> infinity`. So no bound on `Q` holds over the clean pairs, although `Q <= 3^(eps k)` holds over all of them from some `k` on.

**Theorem 3.4 (the four moves).** For every `k >= 0` and every `tau > 0`:

- (i) `h_k(tau') <= 2 h_k(tau)` whenever `tau' > 0` and `abs(tau' - tau) <= 3^(1-k)`, and the factor `2` is sharp: `h_0(1) = 1` and `h_0(4) = 1/2`;
- (ii) `(3/2) h_k(3 tau) <= h_(k+1)(tau) <= 3 h_k(3 tau)`;
- (iii) `(3/8) h_k(tau) <= h_k(4 tau) <= (3/2) h_k(tau)`;
- (iv) `h_(k+1)(tau) >= h_k(tau)`.

Along the chain, `h_(k+1)(tau_(k+1))/h_k(tau_k)` lies in `[9/16, 9/2]`.

**Theorem 3.5 (the average over the scaling).** For every `k >= 0`, `int_1^4 h_k(tau) dtau <= 190`. Hence `leb(G_k(M)) <= 190/M` for every `M > 0`.

**Theorem 3.6 (the moving target).** The upper density of `S` is at least `6/(11 liminf_k h_k(tau_k))`, read as `0` when the liminf is infinite; so it is positive as soon as `liminf_k h_k(tau_k) < infinity`, and no mean over `k` is needed. Contrapositively, if the upper density of `S` is `0`, then for every `M` the chain point `tau_k` lies in `G_k(M)` for all large `k`, where `G_k(M)` has measure at most `190/M` and contains the part in `[1, 4]` of the `3^(1-k)`-neighbourhood of `G_k(2M)`, and where `log_4 tau_(k+1) = log_4 tau_k - log_4 3` modulo `1`.

**Theorem 3.7 (Marstrand at this pair, the lattice identity, almost every phase).**

- (i) At every `tau > 0`, `h_k(tau)` increases with `k` to a limit `h(tau)` in `(0, infinity]`, `int_1^4 h(tau) dtau <= 190`, and `leb(C_3 + tau C_4) >= 1/h(tau)`; so `leb(C_3 + tau C_4) > 0` for almost every `tau in [1, 4]`.
- (ii) At every `tau = 4^m/3^k`, `leb(C_3 + tau C_4) = 3^(-k) card(A_k + B_m) leb(C_3 + C_4)`.
- (iii) For a phase `psi in [1, 4)`, let `tau_k(psi)` be `psi tau_k` brought into `[1, 4)` by a power of `4`, so `tau_k(1) = tau_k`. Then `int_1^4 liminf_k h_k(tau_k(psi)) dpsi <= 760`: for almost every phase the rotated orbit misses the target infinitely often, and the phases with `liminf_k h_k(tau_k(psi)) > M` have measure at most `760/M`.

**Conjecture 3.8.** `liminf_k h_k(tau_k) < infinity`, so the upper density of `S` is positive, at least `6/(11 liminf_k h_k(tau_k))`. The stronger form, a bounded mean of `Q(k, m_k)` over `k <= K`, is what the census suggests: over the `24` chain levels `6 <= k <= 29`, `Q(k, m_k)` lies in `[1.638124, 2.004783]` with mean `1.861840` (Fact 11.2). Its evidence and failure modes close Section 11.

**What is new and what is not.** Lemma 2.1 and the injection are the thread's reading of the Lean proof, and the reflection of Lemma 4.1 is on A367090 on a narrower window. Theorem 3.2 is Theorem C of Glasscock, Moreira and Richter at this pair, restated for the energy rather than the count; its one input is Shmerkin's uniform limit, and the step here is the bridge. The singular scalings behind Theorem 3.3 and the resonance that produces them are those of Nazarov, Peres and Shmerkin, who name this pair. The almost-everywhere half of Theorem 3.7(i) is Marstrand's projection theorem, and the proof of Theorem 3.5 is the energy argument of its potential-theoretic proof, written out for this pair. New here, as far as the sources read here show: the exact bridge of Theorem 3.1; the use of the resonances to make `Q` unbounded on the clean pairs, which closes the route through a uniform bound on `Q`; the four moves; the constants `190`, `760` and `6/11`; the reduction of Theorem 3.6 to a single orbit; the lattice identity of Theorem 3.7(ii); and the census of Section 11.

## The integers

Three facts about the levels come before the continuum: where the sumset is exactly the sum of two levels, how its energy is a correlation of digit counts, and when a pair is only a copy of a smaller one.

**Lemma 4.1 (the clean centres).** The pair `(k, m)` is clean exactly when `3^(k+1) + 5 > 2 4^m` and `4^(m+1) + 5 > 3^(k+1)`, which up to the two `5`s is the window `3/4 < 4^m/3^k < 3/2`. At a clean pair `S meet [0, d] = A_k + B_m`, and `x -> d - x` maps `S meet [0, d]` onto itself. The clean pairs occur at a set of `k` of density `1/2`.

**Proof.** `3^k > (3^k - 1)/2 + (4^m - 1)/3` rearranges to `3^(k+1) + 5 > 2 4^m`, and `4^m > d` to `4^(m+1) + 5 > 3^(k+1)`. At a clean pair a sum `a + b <= d` has `a < 3^k` and `b < 4^m`, since otherwise it would be at least `min(3^k, 4^m) > d`; so `S meet [0, d] = A_k + B_m`. Complementing every digit, `a -> (3^k - 1)/2 - a` maps `A_k` onto itself and `b -> (4^m - 1)/3 - b` maps `B_m` onto itself, and together they send `a + b` to `d - (a + b)`. The window has length `log_4 2 = 1/2` in the variable `m - k log_4 3`, which is equidistributed modulo `1` because `log_4 3` is irrational. □

**Lemma 4.2 (the energy).** `card(A_k + B_m) >= 4^(k+m)/E(k, m) = (d + 1)/Q(k, m)`, and `Q(k, m) >= 1`. Moreover `E(k, m) = sum_t R_A(t) R_B(t)`, where `R_A(t) = 2^(z_3(t))` with `z_3(t)` the number of zero digits of the `k`-digit balanced ternary expansion of `t`, and `R_A(t) = 0` when `abs(t) > (3^k - 1)/2`; and `R_B(t) = 2^(z_4(t))` when `t` has an `m`-digit base-4 expansion with digits in `{-1, 0, 1}`, `z_4(t)` its zero digits, and `R_B(t) = 0` otherwise. Consequently `D(d) >= ((d + 1)/Q(k, m) - 1)/d`, and the upper density of `S` is at least `limsup 1/Q(k, m)` along any infinite family of pairs.

**Proof.** `sum_x r(x) = 2^(k+m)` over the `card(A_k + B_m)` values of `x` with `r(x) > 0`, so Cauchy-Schwarz gives `4^(k+m) <= card(A_k + B_m) E(k, m)`; with `A_k + B_m` inside `[0, d]` the same inequality gives `Q >= 1`. The energy counts the quadruples with `a - a' = b' - b`, so `E = sum_t R_A(t) R_B(t)`, with `R_A(t)` the number of pairs in `A_k^2` with difference `t` and `R_B` likewise, which is even in `t`. A difference of two digit strings from `{0, 1}` is a digit string from `{-1, 0, 1}`, and such a string determines `t` uniquely in base `3`, being the balanced ternary expansion, and in base `4`, the digits `{-1, 0, 1}` being distinct modulo `4`; a nonzero digit of the difference fixes both digits, and a zero digit arises twice. So `R_A(t) = 2^(z_3(t))` and `R_B(t) = 2^(z_4(t))` where the strings exist. Finally `card(S meet [1, d]) >= card(A_k + B_m) - 1`. □

This is the sumset form of the two-base transversality: `Q` bounded says the law of `a - a'` in base `3` and the law of `b - b'` in base `4`, two digit-count weights `2^(z_3)` and `2^(z_4)`, collide no more often than two flat laws on the same range. The two weights are read in multiplicatively independent bases, and Section 12 returns to that obstruction.

**Lemma 4.3 (the gap copies).** When `2 4^m < 3^k + 5`, `A_k + B_m` is the disjoint union of `A_(k-1) + B_m` and its translate by `3^(k-1)`, and `E(k, m) = 2 E(k - 1, m)`. When `3^(k+1) < 4^m + 5`, it is the disjoint union of `A_k + B_(m-1)` and its translate by `4^(m-1)`, and `E(k, m) = 2 E(k, m - 1)`.

**Proof.** `A_k` is `A_(k-1)` together with its translate by `3^(k-1)`, so `r` is the sum of the representation function of the smaller pair and its translate. The two supports are disjoint when `d(k - 1, m) < 3^(k-1)`, which rearranges to `2 4^m < 3^k + 5`, and then the sum of squares doubles. The base-4 case is the same with `B_m = B_(m-1) + {0, 4^(m-1)}` and the condition `d(k, m - 1) < 4^(m-1)`, which is `3^(k+1) < 4^m + 5`. □

At a gap copy `Q` is the smaller pair's value inflated by the gap, `Q(k, m) = Q(k - 1, m) (d(k, m) + 1)/(2 (d(k - 1, m) + 1))` in the first case, so the census of Section 11 reads the copies but keeps them out of its extremes and fits.

## The bridge

**Proof of Theorem 3.1.** Split the first `k` digits off `a` and the first `m` off `b`: `a = 3^(-k) (a_0 + u)` with `a_0 = sum_(l <= k) X_l 3^(k-l)` uniform on `A_k` and `u = sum_(l > k) X_l 3^(k-l)` an independent variable of law `mu`, and `b = 4^(-m) (b_0 + v)` with `b_0` uniform on `B_m` and `v` of law `nu`. Since `tau 4^(-m) = 3^(-k)`, `a + tau b = 3^(-k) (x + w)` with `x = a_0 + b_0` and `w = u + v` in `[0, 1/2 + 1/3] = [0, 5/6]`. So the point lies in the cell `c_x`, and `x` takes the value `x` with probability `r(x)/2^(k+m)`; that is `rho_tau(c_x) = r(x)/2^(k+m)`, whence `sum_x rho_tau(c_x)^2 = E(k, m)/4^(k+m)` and the formula for `Q`. The support `C_3 + tau C_4` is the union over `x in A_k + B_m` of the pieces `3^(-k) (x + C_3 + C_4)`, each nonempty and inside `c_x`, so the cells it meets are exactly the `card(A_k + B_m)` cells `c_x`, and each piece has measure at most `(5/6) 3^(-k)`. For the last claim fix `eps > 0` and a large `x` with `card(S meet [1, x]) < eps x`, and take the chain level `k` with `3^k <= 6x/11 < 3^(k+1)`. Then `d_k < 3^k (1/2 + 4/3) <= x`, so `A_k + B_(m_k)` lies in `S meet [0, x]` and has fewer than `1 + eps x` members, and since `3^(-k) < 11/(2x)`, `leb(C_3 + tau_k C_4) < (5/6)(11/(2x))(1 + eps x) < 5 (eps + 1/x)`. As `eps -> 0` along such `x`, `k -> infinity` and the measure tends to `0`. □

So the fill `card(A_k + B_m)/(d + 1)` is the box count of the continuum sumset `C_3 + tau C_4` at the scale tied to `tau`, and `Q` is its `L^2` sum there. Everything below reads the integer problem through this identity, and the identity pins the scaling: the integers see `C_3 + tau C_4` only at the lattice scalings `4^m/3^k`, and only at the one resolution `3^(-k)` that the scaling fixes. The theorems in print about `C_3 + tau C_4` hold for every scaling or for almost every one, and neither kind singles out the lattice.

## The energy ratio is subpolynomial

**Proof of Theorem 3.2.** First let `tau = 4^m/3^k` lie in `[1, 4)`. The image of `rho_tau` under `y -> 3y` is the measure of Shmerkin's Lemma 7.1 at the point `theta` of his circle with `e^theta = 3 tau/4` or `3 tau`. The separation hypothesis of his Theorem 7.2 holds for both factors with `R = 2`: a nonzero polynomial `P` of degree `n` with coefficients in `{-1, 0, 1}` has `abs(P(1/3)) >= 3^(-n)`, because `3^n P(1/3)` is a nonzero integer by the uniqueness of balanced ternary expansions, and `abs(P(1/4)) >= 4^(-n) >= 3^(-2n)` in the same way in base `4`. Since `D_2 = log_3 2 + 1/2 > 1`, his Theorem 1.11 at `q = 2`, whose limit is uniform over the circle, gives for every `eps > 0` an `n_0` such that the sum of squares of that image over the intervals `[j 2^(-n), (j + 1) 2^(-n))` is at most `2^(-n (1 - eps))` at every `n >= n_0` and every point of the circle. Pulled back by `y -> y/3`, the intervals of length `2^(-n)/3` carry the same sum of squares of `rho_tau`. Choose `n` with `2^(-n)/3` in `[3^(-k), 2 3^(-k))`. A cell meets at most two of these intervals, so its mass squared is at most twice the sum of their squared masses, and each interval meets at most three cells; hence `sum_x rho_tau(c_x)^2 <= 6 2^(-n (1 - eps)) <= 6 (6 3^(-k))^(1 - eps)`. With `d + 1 <= 2 3^k` for `k >= 2`, Theorem 3.1 gives `Q(k, m) <= 72 3^(eps k)`, and running the argument at `eps/2` absorbs the `72` for large `k`.

Now let `tau` lie in `[1/3, 1)`. Split the first digit off `a`: `a = (a_1 + u)/3` with `a_1` uniform on `{0, 1}` and `u` a fresh variable of law `mu`. So `rho_tau` is the average of `rho'` and its translate by `1/3`, where `rho'` is the image of `rho_(3 tau)` under `y -> y/3`. At level `k` the translate moves by exactly `3^(k-1)` cells, and the average of two measures with the same sum of squares over a common cell grid has at most that sum of squares; and `rho'` on the cells of level `k` is `rho_(3 tau)` on the cells of level `k - 1`. So `E(k, m)/4^(k+m) <= E(k - 1, m)/4^(k-1+m)`, that is `E(k, m) <= 4 E(k - 1, m)`, and since `d(k, m) + 1 <= 3 (d(k - 1, m) + 1)`, `Q(k, m) <= 3 Q(k - 1, m)` with `4^m/3^(k-1) = 3 tau` in `[1, 3)`. The first case bounds that.

For the count, given a large `x` take the chain level `k` with `d_k <= x < d_(k+1)`; the `d_k` increase, since `d_(k+1) + 5/6 = 3^(k+1) (1/2 + tau_(k+1)/3) >= (5/2) 3^k > (11/6) 3^k > d_k + 5/6`. Then `x < 6 3^k` and `d_k + 1 >= (5/6) 3^k > x/8`, so `card(S meet [1, x]) >= (d_k + 1)/Q(k, m_k) - 1 >= x^(1 - 2 eps)` for large `x`. □

This is Theorem C of Glasscock, Moreira and Richter at this pair, whose proof also runs through Shmerkin's uniformity; the form here bounds the energy and not only the count, and it is sharp in the only sense available, by the next section.

## The energy ratio is unbounded

**Proof of Theorem 3.3.** In the symmetric coordinates of Nazarov, Peres and Shmerkin their Cantor measures of ratios `1/3` and `1/4` are affine images of `mu` and `nu`, and their convolution at scaling `lambda` is an affine image of `rho_tau` with `tau = 4/(3 lambda)`. Their resonance `abs(lambda 4^n - 3^m) < 1/4`, multiplied by `3 tau`, reads `abs(tau 3^k - 4^(m')) < 3 tau/4` at `(k, m') = (m + 1, n + 1)`. The proof of their Theorem 4.1 finds the Fourier transform away from `0` at every resonance, so at a `tau` with infinitely many resonances the Fourier transform of `rho_tau` does not tend to `0` and, by the Riemann-Lebesgue lemma, `rho_tau` has no density; their theorem makes these `tau` a dense `G_delta` and `rho_tau` singular there. Only the absence of an `L^2` density is used. Fix such a `tau` in `(1, 5/4)`. If `3^k sum_x rho(c_x)^2` stayed bounded along a sequence of `k` for a probability measure `rho`, the cell averages `3^k sum_x rho(c_x) 1_(c_x)` would be bounded in `L^2`, a subsequence would converge weakly to some `f` in `L^2`, and since the cells shrink, `rho = f dx`; so `h_k(tau) -> infinity`, monotonically by move (iv). At a resonant pair `(k, m)`, `tau_(k,m) = 4^m/3^k` is within `3 tau 3^(-k)/4 < 3^(-k)` of `tau`. Replacing `a + tau b` by `a + tau_(k,m) b` moves every point by at most `3^(-k)/3`, as `b <= 1/3`, and all in one direction, so the mass of each cell of `rho_(tau_(k,m))` stays in its cell or moves to the next one, and by the split of move (i), proved in Section 8, `sum_x rho_tau(c_x)^2 <= 2 sum_x rho_(tau_(k,m))(c_x)^2`. With `d + 1 >= 3^k/2`, Theorem 3.1 gives `Q(k, m) >= h_k(tau)/4 -> infinity`. As `tau_(k,m) -> tau` inside `(1, 5/4)`, which lies in the clean window `(3/4, 3/2)` with room for the two `5`s, these pairs are clean from some `k` on. □

**Proposition 7.1 (the set route).** If some `tau in [1, 4)` has `abs(tau 3^k - 4^m) < 3 tau/4` for infinitely many pairs `(k, m)` and `leb(C_3 + tau C_4) > 0`, then `S` has positive upper density. At every such `tau` the measure `rho_tau` is singular.

**Proof.** At a resonant pair `abs(tau - tau_(k,m)) < 3 tau 3^(-k)/4 < 3^(1-k)`, and `b <= 1/3`, so every point `a + tau b` of `C_3 + tau C_4` sits less than one cell from `a + tau_(k,m) b`, on one fixed side, and that point lies in one of the `card(A_k + B_m)` cells of Theorem 3.1, at most `5/6` of a cell from its left end. So every cell meeting `C_3 + tau C_4` is one of those cells or its neighbour on that side, and `card(A_k + B_m)` is at least half the number `N_k` of cells meeting `C_3 + tau C_4`. The union of those cells covers the set, so `N_k 3^(-k) >= leb(C_3 + tau C_4)`, and `D(d) >= (card(A_k + B_m) - 1)/d >= leb(C_3 + tau C_4)/4 - 1/d` with `d + 1 <= 2 3^k`. Singularity is Nazarov, Peres and Shmerkin's Theorem 4.1 again, since `tau` has infinitely many resonances. □

So the resonant scalings are the ones whose energy blows up, and a set route through them needs a singular measure with a support of positive length, which nothing here rules out or supplies.

## The four moves

Each move compares two cell energies by one elementary inequality on sums of squares. Two are used repeatedly: moving mass by less than a cell, and averaging two copies.

**Proof of Theorem 3.4.** (i) Since `b <= 1/3`, replacing `tau` by `tau'` with `abs(tau' - tau) <= 3^(1-k)` moves every point `a + tau b` by at most `3^(-k)`, one cell, and all in one direction, say up. The mass `rho_tau(c_x)` then splits into a part `s_x` that stays in `c_x` and a part `v_x` that moves to `c_(x+1)`, so `rho_(tau')(c_x) = s_x + v_(x-1)` and `sum_x (s_x + v_(x-1))^2 <= 2 sum_x (s_x^2 + v_(x-1)^2) <= 2 sum_x rho_tau(c_x)^2`, as `s_x^2 + v_x^2 <= (s_x + v_x)^2`. For sharpness, at `k = 0` and `tau = 1` the whole support `[0, 5/6]` lies in `c_0`, so `h_0(1) = 1`, while at `tau = 4`, `4 b = Y_1 + b'` with `b'` in `[0, 1/3]`, so the mass splits evenly between `c_0` and `c_1` and `h_0(4) = 1/2`; the scalings are `3 = 3^(1-0)` apart.

(ii) Split the first digit off `a`, as in the proof of Theorem 3.2: `rho_tau` is the average of `rho'` and its translate by `1/3`, where `rho'`, the image of `rho_(3 tau)` under `y -> y/3`, carries on the cells of level `k + 1` exactly the masses of `rho_(3 tau)` on the cells of level `k`, and the translate moves by `3^k` whole cells of level `k + 1`. For nonnegative `p` and `q`, `((p + q)/2)^2` lies between `(p^2 + q^2)/4` and `(p^2 + q^2)/2`; summing, `3^(-k-1) h_(k+1)(tau)` lies between `3^(-k) h_k(3 tau)/2` and `3^(-k) h_k(3 tau)`.

(iii) Split the first digit off `b`: `b = (b_1 + v)/4` with `b_1` uniform on `{0, 1}` and `v` of law `nu`, so `rho_(4 tau)` is the average of `rho_tau` and its translate by `tau`. A translate by any amount changes the sum of squares over the cells by a factor in `[1/2, 2]`: a whole number of cells changes nothing, and the fractional part is a move of less than one cell, handled by the split of (i) in both directions. With `p` the masses of `rho_tau` and `q` those of the translate, `sum ((p + q)/2)^2 <= (sum p^2 + sum q^2)/2 <= (3/2) sum p^2` and `>= (sum p^2 + sum q^2)/4 >= (3/8) sum p^2`.

(iv) A cell of level `k` is the union of three cells of level `k + 1`, and `(p_1 + p_2 + p_3)^2 <= 3 (p_1^2 + p_2^2 + p_3^2)`, so `3^(-k) h_k(tau) <= 3 3^(-k-1) h_(k+1)(tau)`.

For the chain, `3 tau_(k+1) = 4^(m_(k+1))/3^k` and `tau_k = 4^(m_k)/3^k` differ by a power of `4`, and with `3 tau_(k+1)` in `[3, 12)` and `tau_k` in `[1, 4)` that power is `1` or `4`. By (ii), `h_(k+1)(tau_(k+1))` lies between `(3/2) h_k(3 tau_(k+1))` and `3 h_k(3 tau_(k+1))`; when `3 tau_(k+1) = tau_k` the ratio lies in `[3/2, 3]`, and when `3 tau_(k+1) = 4 tau_k`, (iii) puts it in `[(3/2)(3/8), 3 (3/2)] = [9/16, 9/2]`. □

So one step of the chain changes the orbit value by a bounded factor, and the chain cannot jump from low ground to arbitrarily high ground in one step; what it can do is climb slowly, and that is what the conjecture denies.

## The average over the scaling

The bound `190` is the energy argument of the potential-theoretic proof of Marstrand's projection theorem, written out with the digit tails of this pair.

**Lemma 9.1 (the digit tails).** With `a, a'` independent of law `mu`, `b, b'` independent of law `nu`, and `alpha = log_3 2`, `P(abs(a - a') < s) <= (6s)^alpha` and `P(abs(b - b') < s) <= (6s)^(1/2)` for every `s > 0`.

**Proof.** If the first digit where `a` and `a'` differ is at place `j`, then `abs(a - a') >= 3^(-j) - sum_(l > j) 3^(-l) = 3^(-j)/2`, and that first difference is at place `j` or later with probability `2^(1-j)`. So `abs(a - a') < s` forces `3^(-j) < 2s` at the first differing place, and with `j_0` the least `j` with `3^(-j) < 2s` the probability is at most `2^(1 - j_0) = 2 (3^(-j_0))^alpha < 2 (2s)^alpha = (6s)^alpha`, as `2^(1/alpha) = 3`. In base `4` a first difference at place `j` gives `abs(b - b') >= 4^(-j) - sum_(l > j) 4^(-l) = (2/3) 4^(-j)`, and the same count gives `2 (3s/2)^(1/2) = (6s)^(1/2)`. □

**Proof of Theorem 3.5.** Write `ell = 3^(-k)`. Two independent points of `rho_tau` in one cell are less than `ell` apart, so `sum_x rho_tau(c_x)^2 <= P(abs(a - a' + tau (b - b')) < ell)`, with `a, a', b, b'` as in Lemma 9.1. Integrate over `tau in [1, 4]` and exchange: for fixed `b != b'` the `tau` in `[1, 4]` with `abs(a - a' + tau (b - b')) < ell` form an interval of length at most `min(3, 2 ell/abs(b - b'))`, empty unless `abs(a - a') < 4 abs(b - b') + ell`, and `b = b'` has probability `0`. By independence and Lemma 9.1,

```
int_1^4 sum_x rho_tau(c_x)^2 dtau <= E[ min(3, 2 ell/abs(b - b')) (6 (4 abs(b - b') + ell))^alpha ] .
```

Where `abs(b - b') >= ell`, `4 abs(b - b') + ell <= 5 abs(b - b')` and the integrand is at most `2 30^alpha ell abs(b - b')^(alpha - 1)`. Where `abs(b - b') < ell` it is at most `3 (30 ell)^alpha`, on an event of probability at most `(6 ell)^(1/2)`. So the integral is at most `2 30^alpha ell E(abs(b - b')^(alpha - 1)) + 3 30^alpha 6^(1/2) ell^(alpha + 1/2)`. Put `gamma = 1/(2 - 2 alpha)`; Lemma 9.1 gives `P(abs(b - b')^(alpha - 1) > t) <= min(1, 6^(1/2) t^(-gamma))`, and integrating in `t`, `E(abs(b - b')^(alpha - 1)) <= 6^(1/(2 gamma)) gamma/(gamma - 1)`, finite exactly because `gamma > 1`, which is `alpha + 1/2 > 1`, the dimensions adding past `1`. The constants are `E(abs(b - b')^(alpha - 1)) <= 7.398167` and `30^alpha <= 8.549875`, and with `ell^(alpha + 1/2) <= ell` the total is at most `189.34 ell` (Fact 11.4). Multiplying by `3^k = 1/ell`, `int_1^4 h_k <= 190`, and Chebyshev's inequality gives `leb(G_k(M)) <= 190/M`. □

The constant sits far above the readings: the curves of the figure, `h_k` computed with the tail of `nu` replaced by its mean and integrated by the trapezoid rule, integrate to between `2` and `4` at every `k <= 6`, which its binary asserts, and the proxy of Fact 11.3 reads between `3.93` and `4.58` at `k = 8..17`. What matters here is that it is a constant, the same at every level.

## The moving target

**Proof of Theorem 3.6.** On the chain `d_k + 1 = 3^k (1/2 + tau_k/3) + 1/6`, so Theorem 3.1 gives `Q(k, m_k) = (d_k + 1) 3^(-k) h_k(tau_k) <= (11/6 + 3^(-k)/6) h_k(tau_k)`, as `tau_k < 4`. The chain is an infinite family of pairs, so by Lemma 4.2 the upper density of `S` is at least `limsup_k 1/Q(k, m_k) >= limsup_k 6/((11 + 3^(-k)) h_k(tau_k)) = 6/(11 liminf_k h_k(tau_k))`. If the upper density is `0`, the liminf is infinite, so for every `M` the value `h_k(tau_k)` exceeds `M` for all large `k`, which is `tau_k in G_k(M)`; Theorem 3.5 bounds its measure. If `tau in [1, 4]` lies within `3^(1-k)` of some `tau'` in `G_k(2M)`, move (i) gives `2M < h_k(tau') <= 2 h_k(tau)`, so `tau` lies in `G_k(M)`. The rotation is the chain step of Theorem 3.4: `tau_(k+1) = 4^j tau_k/3` with `j` in `{0, 1}`. □

So zero upper density is a shrinking-target statement about one orbit: the rotation by `-log_4 3` on the circle `[1, 4)`, started at `1`, would have to lie in `G_k(M)` at every late step `k`, for every `M`, while `G_k(M)` has measure at most `190/M` and holds an interval of radius `3^(1-k)` around every point where `h_k > 2M`, so the orbit cannot slip through a thin spike. The orbit does enter such targets infinitely often: the unbounded family of Theorem 3.3 has its scalings `tau_(k,m)` converging to a `tau` in `(1, 5/4)`, inside `[1, 4)`, so its pairs are chain pairs and `limsup_k h_k(tau_k) = infinity`. Whether it also leaves them infinitely often is the question.

**Proof of Theorem 3.7.** (i) By move (iv), `h_k(tau)` is nondecreasing in `k`, and `h_0(tau) > 0`; monotone convergence and Theorem 3.5 give `int_1^4 h <= 190`. Let `N_k` be the number of cells of level `k` that meet `C_3 + tau C_4` and `U_k` their union. Those cells carry all the mass, so Cauchy-Schwarz gives `1 <= N_k sum_x rho_tau(c_x)^2`, that is `leb(U_k) = N_k 3^(-k) >= 1/h_k(tau) >= 1/h(tau)`. A cell of level `k + 1` that meets the set lies in a cell of level `k` that meets it, so the `U_k` decrease, and their intersection lies within `3^(-k)` of the compact set `C_3 + tau C_4` for every `k`, hence in it. So `leb(C_3 + tau C_4) >= lim_k leb(U_k) >= 1/h(tau)`, positive wherever `h` is finite, which is almost everywhere on `[1, 4]`. (ii) At `tau = 4^m/3^k` the pieces `3^(-k) (x + C_3 + C_4)`, `x in A_k + B_m`, lie in distinct cells by Theorem 3.1, and each has measure `3^(-k) leb(C_3 + C_4)`. (iii) For fixed `k`, `psi -> tau_k(psi)` is a rotation of the circle `[1, 4)` in the coordinate `log_4`, piecewise `tau = 4^(-j) psi tau_k`, with `dpsi/dtau = psi/tau <= 4`. So `int_1^4 h_k(tau_k(psi)) dpsi <= 4 int_1^4 h_k(tau) dtau <= 760`, and Fatou's lemma gives `int_1^4 liminf_k h_k(tau_k(psi)) dpsi <= 760`; Chebyshev's inequality gives the last claim. □

Part (i) is Marstrand's theorem for this pair, with an explicit density: `leb(C_3 + tau C_4) >= 1/h(tau)` and `int h <= 190`. Part (ii) puts the one number `leb(C_3 + C_4)` in front of every lattice scaling, so the continuum sumset at the scalings the integers see is the integer sumset times a fixed factor; whether that factor is positive is the case `r = 3`, `s = 4`, `X = C_3`, `Y = C_4` of Hochman's question. Part (iii) is the moving target at a random phase, and the phase of `S` is `psi = 1`, a single point, which no almost-every statement reaches.

## The census

Every number below is printed by `lab/rs/sumset-density`, one verb per fact, and the integer objects are those of `mrlyrs::num::sumset`. Densities are truncated to six places, energy ratios rounded up, and the ends of an interval of ratios printed truncated below and rounded up above.

**Fact 11.1 (the density to `3^22`).** `card(S meet [1, 3^22]) = 26666749554` and `D(3^22) = 0.849772`. Table 1 gives `D(3^k)` and the extremes of `D` over each window `[3^k, 3^(k+1))`. `D(4^m)` at `m = 3..17` reads `0.968750, 0.843750, 0.860351, 0.897460, 0.859313, 0.791305, 0.837238, 0.868845, 0.806823, 0.783585, 0.838184, 0.875988, 0.785523, 0.793552, 0.845272`. Over all of `[1, 3^22]` the minimum of `D` is `0.763391`, at `x = 3^15 - 1`, and the maximum is `1`, at every `x <= 61`; the least of the dyadic maxima over `[2^j, 2^(j+1))`, `j = 8..33`, is `0.841760`. The deepest readings sit where `4^m/3^k` is nearest `1`, at `3^k` when the ratio is above `1` and at `4^m` when it is below: `0.778472` at `3^10` with `4^8/3^10 = 1.109858`, `0.763392` at `3^15` with `4^12/3^15 = 1.169234`, `0.767893` at `3^20` with `4^16/3^20 = 1.231785`, and `0.785523` at `4^15` with `4^15/3^19 = 0.923839`, the ratios rounded up; that is the gap of Lemma 2.1 at each near coincidence, and nothing deeper. The falsifier set in advance, window maxima decaying like a power, does not appear: the maximum reads `0.913419` at `k = 5`, `0.910650` at `k = 18` and `0.872186` at `k = 21`, and the proved zero lower density is invisible at this height. Controls: the bit array matches a double loop over `A x B` to `3^13` and the other shift order to `3^17`, its first `58` non-members are the first `58` terms of A367090, and the reflection `x -> d - x` fixes `S meet [0, d]` at every clean centre below `3^17` and at no mixed centre with `d >= 449`. Generator: `lab/rs/sumset-density`, verbs `density 22` and `control`.

| `k` | `D(3^k)` | max on `[3^k, 3^(k+1))` | min on `[3^k, 3^(k+1))` |
| ---: | ---: | ---: | ---: |
| `5` | `0.835390` | `0.913419` | `0.835390` |
| `6` | `0.858710` | `0.903768` | `0.852729` |
| `7` | `0.887517` | `0.912038` | `0.887517` |
| `8` | `0.908855` | `0.931596` | `0.858945` |
| `9` | `0.864959` | `0.913781` | `0.778468` |
| `10` | `0.778472` | `0.875566` | `0.778472` |
| `11` | `0.837186` | `0.875469` | `0.822506` |
| `12` | `0.858264` | `0.881621` | `0.858264` |
| `13` | `0.874244` | `0.908274` | `0.806430` |
| `14` | `0.814704` | `0.885045` | `0.763391` |
| `15` | `0.763392` | `0.865671` | `0.763392` |
| `16` | `0.831183` | `0.882855` | `0.815887` |
| `17` | `0.858962` | `0.886340` | `0.858962` |
| `18` | `0.881342` | `0.910650` | `0.785230` |
| `19` | `0.792352` | `0.874408` | `0.767893` |
| `20` | `0.767893` | `0.865858` | `0.767875` |
| `21` | `0.831191` | `0.872186` | `0.818358` |

**Table 1.** The density of `S` at the powers of `3` and its extremes over the windows between them. The least window maximum is `0.865671`, at `k = 15`.

**Fact 11.2 (the energy ladder to `k = 29`).** Over the `43` pairs with `6 <= k <= 29` and `1/3 < 4^m/3^k < 4`, twelve are gap copies in the sense of Lemma 4.3: `(6, 4), (7, 5), (11, 8), (12, 9), (16, 12), (21, 16), (26, 20)` and the chain levels `(9, 8), (14, 12), (19, 16), (24, 20), (28, 23)`. Over the `31` others `Q` lies in `[1.638124, 2.004783]`, least at `(6, 5)`, where `Q = 858849/524288`, and largest at `(26, 21)`; so no `Q` passes `2.07` to `k = 29`, and the largest reading of all, `2.060586` at `(16, 12)`, is a copy. The whole chain, `24` levels with its copies, lies in the same interval with mean `1.861840` (Table 2). Least squares of `log_3 Q` on `k` give `eta = 0.003915` with standard error `0.001243` over the `31`, and `eta = -0.001109` with standard error `0.001254` over the `25` with `k >= 11`: past `k = 11` the slope is zero within one standard error. Scored in units of `Q` on those `25`, the power model `3^(eta k)` has `eta = -0.001093` and rms `0.065794`, and the model `a + b 3^(-s k)`, `s = log_3 2 - 1/2 = 0.130930`, has rms `0.065127` with `a = 1.869317` and `b = 0.282604`, a slow decline rather than a rise to a limit; over all `31` the same model has `b = -0.606850`, so the sign depends on the window and the census separates neither model. On the `27` pairs with `6 <= k <= 22`, `4^m` within a factor `3` of `3^k` and `d <= 3^22`, the Cauchy-Schwarz bound `1/Q` alone puts `D(d)` at or above `0.485298`, against fills `card(S meet [0, d])/(d + 1)` between `0.834213` and `0.928391`. Generator: `lab/rs/sumset-density`, verbs `ladder 29` and `energy 22`.

| `k` | `m_k` | `tau_k` | pair | `Q(k, m_k)` |
| ---: | ---: | ---: | :--- | ---: |
| `6` | `5` | `1.404664` | clean | `1.638125` |
| `7` | `6` | `1.872886` | mixed | `1.724517` |
| `8` | `7` | `2.497181` | mixed | `1.646953` |
| `9` | `8` | `3.329574` | copy | `1.700067` |
| `10` | `8` | `1.109858` | clean | `1.761631` |
| `11` | `9` | `1.479811` | clean | `1.941866` |
| `12` | `10` | `1.973081` | mixed | `1.965842` |
| `13` | `11` | `2.630775` | mixed | `1.878861` |
| `14` | `12` | `3.507700` | copy | `1.954950` |
| `15` | `12` | `1.169234` | clean | `1.940363` |
| `16` | `13` | `1.558978` | mixed | `1.987226` |
| `17` | `14` | `2.078637` | mixed | `1.860206` |
| `18` | `15` | `2.771516` | mixed | `1.803557` |
| `19` | `16` | `3.695354` | copy | `1.906659` |
| `20` | `16` | `1.231785` | clean | `1.895327` |
| `21` | `17` | `1.642380` | mixed | `1.959133` |
| `22` | `18` | `2.189840` | mixed | `1.861153` |
| `23` | `19` | `2.919786` | mixed | `1.822641` |
| `24` | `20` | `3.893048` | copy | `1.942757` |
| `25` | `20` | `1.297683` | clean | `1.925912` |
| `26` | `21` | `1.730244` | mixed | `2.004783` |
| `27` | `22` | `2.306992` | mixed | `1.911416` |
| `28` | `23` | `3.075989` | copy | `1.832435` |
| `29` | `23` | `1.025330` | clean | `1.817784` |

**Table 2.** The chain: at each level `k` the one pair `(k, m_k)` with `tau_k = 4^(m_k)/3^k` in `[1, 4)`, whether it is clean, mixed or a gap copy, and its energy ratio, rounded up. By Theorem 3.1, `h_k(tau_k) = Q(k, m_k) 3^k/(d_k + 1)`.

**Fact 11.3 (where the orbit sits, a proxy).** For each `k = 8..17` on the chain, the energy of the atoms `a + sigma b`, `a` in `A_k` and `b` in `B_(m_k)`, is read at `4096` scalings `tau = sigma tau_k` spread evenly in `log tau` over `[1, 4)`, once with a nearest-integer window and once with the tent `(1 - abs(Delta))_+`, `Delta` the difference of two atoms. Both equal `E(k, m_k)` at `sigma = 1` and are a proxy for `h_k` elsewhere, not `h_k` itself, so every rank and ratio here is the proxy's. The integral of the window reading over `[1, 4]` is `3.937434`, `4.318193` and `4.572550` at `k = 8, 12, 17`, far under the `190` proved for `h_k`. Its maximum over the period sits at the grid point `1.053504`, next to the lattice point `4^4/3^5`, at `k = 8` and every `k >= 11`, and at `1.404673`, next to `4^5/3^6`, at `k = 9, 10`: the large values sit at the low lattice points, where the two bases coincide early. Among the grid points within `1/50` of `tau_k` in `log_4 tau`, the share below the orbit value averages `0.8110` over the ten levels under both kernels, between `0.7055` and `0.9202` under the window, and the orbit value exceeds their mean by a factor between `1.0090` and `1.0525` under the window, largest at `k = 12` and `1.0180` at `k = 17`, and between `1.0063` and `1.0512` under the tent. The orbit sits a few percent above its neighbourhood, and the excess does not grow over this range. Generator: `lab/rs/sumset-density`, verb `phases 17 4096`.

**Fact 11.4 (the constants of Theorem 3.5).** `E(abs(b - b')^(alpha - 1)) <= 7.398167` and `30^alpha <= 8.549875`, each the exact value rounded up, and the exact total `30^alpha (2 6^(1/(2 gamma)) gamma/(gamma - 1) + 3 6^(1/2))` rounded up is `189.335292`; from the two rounded constants the total is below `189.34`. Generator: `lab/rs/sumset-density`, verb `constants`.

**Conjecture 3.8, weighed.** For it: past `k = 11` the chain readings are flat within one standard error; the orbit value sits only a few percent above its neighbourhood, with no growth over `k = 8..17`; and a scaling with infinitely many resonances meets the chain at only `O(log K)` of the levels `k <= K`, because two resonances `(k_1, m_1)` and `(k_2, m_2)` of one `tau` make `abs(4^(m_2 - m_1)/3^(k_2 - k_1) - 1)` smaller than about `3^(-k_1)`, while `abs(4^j - 3^i) >= 1` keeps it at least `3^(-i)`, so `k_2 >= 2 k_1` up to a bounded shift. Against it: the orbit does climb, `limsup_k h_k(tau_k) = infinity` by Section 10, and a factor `3^(eta k)` in `Q` with `eta` below `0.01` is invisible to `k = 29`. It fails if the local excess of `h_k(tau_k)` over its neighbourhood keeps growing with `k`, or if the chain readings pass `2.1` and keep climbing.

## Open problems

The upper density of `S` stays open, and this paper leaves it at one orbit: whether `liminf_k h_k(tau_k) < infinity` at the single phase `psi = 1`, Conjecture 3.8. Every bound here averages, over the scaling in Theorem 3.5 or over the phase in Theorem 3.7, the resonances of Theorem 3.3 show that no bound holds uniformly, and so a proof has to use the arithmetic of the one orbit `tau_k = 4^(m_k)/3^k`, where `Q` is the correlation of the digit weights `2^(z_3)` and `2^(z_4)`, read in two independent bases. The second problem is whether `leb(C_3 + C_4) > 0`, the case `r = 3`, `s = 4` of Hochman's question: by Theorem 3.7(ii) that one number scales the continuum sumset at every lattice scaling, and by Proposition 7.1 a resonant scaling whose sumset has positive length would give positive upper density, but whether any resonant scaling has one is open, the measure on it being singular. The rest is size: the constant `190` sits far above the integrals near `4` that the readings show, `6/11` is not optimized, the census stops at `k = 29` for the energies and at `3^22` for the density, and the proved zero lower density never shows below `3^22`, where the density stays above `0.76`.

## Reproducibility

Every computed number of Sections 1 and 11, and the census reading cited in Section 2, is printed by the study `research/lab/rs/sumset-density`, a crate of the root workspace, run from the repository root as `CARGO_BUILD_JOBS=4 cargo run --release -p sumset-density -- <verb>`. `density 22` prints Fact 11.1, Table 1 and the counts of the introduction in `13` seconds on a `4` GB bit array, and `density 17` a smaller run in `0.09` seconds; `control` prints the controls of Fact 11.1 in `0.08` seconds; `energy 22` prints the `27` pairs with `d <= 3^22`, their fills and their Cauchy-Schwarz bounds, in `39` seconds; `ladder 29 energies.tsv` prints Fact 11.2 and Table 2, the energies computed once in `139` to `178` seconds on `8` threads and cached in the named file, and in under a second from a full cache; `phases 17 4096` prints Fact 11.3 in `47` seconds on `8` threads; `constants` prints Fact 11.4 at once. `cargo test --release -p sumset-density` runs `11` tests, which check the energy against the histogram of the representation function, the chunked energy of the ladder against the digit-string energy at every `k <= 11`, `m <= 14`, the gap copies of Lemma 4.3, and both phase kernels against `E(k, m)` at the lattice. The integer objects live in `mrlyrs::num::sumset`: `Sumset` holds `S` as one bit per integer to `3^20`, and `Pair` carries a pair's `largest`, `gap`, `clean`, `copy`, `scale`, `energy` and `ratio`, the energy summed over the `3^m` digit strings of Lemma 4.2. The demo [three plus four](/demos/sumset/) runs the same crate in the browser to `3^16`: `S` below `x` as a zoomable strip, `D(x)` dipping at every gap, and `Q` along the pairs. The figure is `bash scripts/figures.sh paper-erdos-125-upper-density`, under a second a theme. Its binary computes `h_k(tau)` for `k <= 6` at `3073` scalings from the distribution function of `mu` and the atoms of `nu` on `ceil(log_4(tau 3^k)) + 7` digits, the rest of `nu` replaced by its mean, which is exact at the lattice scalings; it asserts the integral of each curve over `[1, 4]` between `2` and `4`, and `h_k(tau_k) = 3^k E(k, m_k)/4^(k + m_k)` against `Pair::energy` at every chain level `k <= 6`, an independent check of Theorem 3.1 from the continuum side; it also asserts move (iv) at every sample, the chain steps inside `[9/16, 9/2]`, `h_0(1) = 1`, `h_0(4) = 1/2`, and `Q(6, 5) = 858849/524288` from `Pair::ratio`.

## References

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