ford-horocycle

• The Ford circles read as horocycles of the modular surface, the closed horocycle Im z = h against the Farey stack, the unfolding that turns a horocycle integral into the stack's smoothed novelty, and the census of the Ford circles whose tangency point has every partial quotient at most m, for apollonian, ## The horocycle.
• crossings takes every reduced a/b in [0, 1) with b <= bound, completes (a, b) to a matrix of SL(2, Z) with bottom row (b, d), maps five points of the line Im z = 1 by it in exact rationals, one of them -d/b + i, and asserts that every image lies on the circle of centre (a/b, 1/(2 b^2)) and radius 1/(2 b^2) at height at most 1/b^2, with equality at x = -d/b. It then counts, at the heights h = Q^(-2), the Ford circles whose closed disc the line Im z = h meets, 0 <= h <= 1/b^2, and asserts the count is sum_(b <= Q) phi(b); the open-disc count is asserted to be sum_(b < Q) phi(b), and the closed-disc count at h = 1/(2 Q^2) to be sum_(b <= floor(Q sqrt 2)) phi(b).
• bridge takes a test function f supported on [1, 2], the C^infinity bump exp(4 - 1/((u-1)(2-u))) and the C^2 bump 64 (u-1)^3 (2-u)^3 of lab/py/smoothed-novelty, forms the incomplete Eisenstein series psi_f(z) = sum f(Im(gamma z)) over the cosets Gamma_infinity \ SL(2, Z), and integrates it over the closed horocycle x + ih, 0 <= x < 1, term by term: each coset with bottom row (c, d) contributes on abs(x + d/c) <= sqrt(h - c^2 h^2)/c, split at the inner hole where Im(gamma z) > 2, by Gauss-Legendre on each piece. It compares the sum with h sum_(c <= h^(-1/2)) phi(c) g(c h^(1/2)), g(t) = int_R f(1/(t^2 (1 + v^2))) dv by the same quadrature on the interval where the integrand is nonzero, and prints both, their gap, the main term (3/pi) int f(w) w^(-2) dw, the error E and E/h^(3/4), at h = 4^(-k), k = 2..kmax.
• census walks the Stern-Brocot tree of (0, 1) from 1/2, each node carrying its two neighbours, the direction of its last step and the length of its last run, and keeps a child only when its denominator is at most 2^jmax and its run length at most m; the count N_m(Q) of nodes of denominator at most Q, the node 0/1 added, is read off a histogram. It first walks with no run bound to 2^jcontrol and asserts N(2^j) = sum_(b <= 2^j) phi(b) at every j; then, at m = 2 and m = 3 to 2^jcheck, asserts that the walk, the test "one of the two continued fraction expansions of a/b has every quotient at most m", and the rule "a_1, ..., a_(n-1) <= m and a_n <= m + 1 on the expansion with a_n >= 2" pick the same fractions, and at every node that the mediant with its younger neighbour stays in the family while the mediant with its older neighbour stays exactly when a_n <= m. Then it prints, per octave Q = 2^j, N_m(Q), Q^(2 delta_m), their ratio, the octave exponent log2(N(2Q)/N(Q)), that exponent minus 2 delta_m, the two-octave exponent log4(N(4Q)/N(Q)), and the least and greatest ratio on the quarter octaves of [j, j + 1), with delta_2 = 0.5312805062772051416 and delta_3 = 0.705660908028 the pressure zeros of beneath, ### The two dimensions of a run-length rule.
• The totients are sieved in numpy; the walk and the quotient tests are exact integers; the horocycle images are exact rationals.

RUN

• uv run python mrlyprod/research/lab/py/ford-horocycle/ford_horocycle.py crossings, 0.1 seconds.
• uv run python mrlyprod/research/lab/py/ford-horocycle/ford_horocycle.py bridge, 21 seconds, 16 of them the two k = 10 rows.
• uv run python mrlyprod/research/lab/py/ford-horocycle/ford_horocycle.py census, 28 seconds, 19 of them the m = 2 walk to 2^24.
• uv run python mrlyprod/research/lab/py/ford-horocycle/ford_horocycle.py all, 49 seconds.
• --bound 60, --kmax 10, --nodes 200, --jmax2 24, --jmax3 18, --jcontrol 10 and --jcheck 9 are the dials.
• Prints only, writes nothing, touches no network.

WITNESSES

• The lines of apollonian, ## The horocycle.
• crossings: 5510 exact image points on the 1102 Ford circles of b <= 60, every one on the Ford circle over a/b with top point 1/b^2 at x = -d/b; closed-disc counts 1, 2, 4, 32, 324, 542 at Q = 1, 2, 3, 10, 32, 42 against sum_(b <= Q) phi(b), open-disc counts 0, 1, 2, 28, 308, 530 against sum_(b < Q) phi(b), and 1, 2, 6, 64, 628, 1086 at h = 1/(2 Q^2) against sum_(b <= floor(Q sqrt 2)) phi(b).
• bridge: the horocycle integral and h sum phi(c) g(c h^(1/2)) agree to 7.7e-14 or better at every h = 4^(-k), k = 2..10, on both bumps, 318453 cosets contributing at k = 10; the main terms are 0.167199591457466 and 0.201589480296982; E/h^(3/4) stays between -0.27 and +0.46 on the smooth bump and between -0.23 and +0.41 on the C^2 bump.
• census: the unbounded walk reads sum_(b <= Q) phi(b) at every Q = 2^j to 2^10, 318964 at the top; at b <= 2^9 the three membership tests agree on 961 fractions at m = 2 and 5118 at m = 3, the mediant with the younger neighbour stays in every case, 960 and 5117, and the mediant with the older neighbour stays in 584 and 4026, exactly the nodes with a_n <= m.
• census, m = 2: N_2(2^24) = 59104562, N_2/Q^(2 delta_2) between 1.2272 and 1.2504 on the quarter octaves of [22, 23) and between 1.2301 and 1.2521 on [23, 24); octave exponents 1.0431 and 1.0750 at j = 22, 23, octave exponent minus 2 delta_2 reading +0.0475, -0.0480, +0.0477, -0.0453, +0.0405, -0.0336, +0.0265, -0.0194, +0.0124 at j = 15 to 23, two-octave exponents between 1.0590 and 1.0661 from j = 15 to j = 22, against 2 delta_2 = 1.0625610125544.
• census, m = 3: N_3(2^18) = 34078973, N_3/Q^(2 delta_3) between 0.76762 and 0.76775 on the quarter octaves of [17, 18), octave exponents 1.4113, 1.4114, 1.4112 at j = 15, 16, 17, two-octave exponents 1.4114, 1.4113 at j = 15, 16, against 2 delta_3 = 1.4113218160560.