The L∞ three-torus bound g_∞ ≤ 2^d+1 is sharp for d = 3: nine distances attained #
The repo's SimDirichlet.nnDist_count_unconditional proves the sup-norm bound g_∞ ≤ 2^d + 1 in
all
dimensions (Shutov), i.e. ≤ 9 for d = 3. Haynes–Marklof–Ramirez
(arXiv:2010.08842, Higher dimensional gap theorems for the
maximum metric, IJNT 17 (2021)) prove this bound is sharp for d ≤ 3, and give an explicit
d = 3 example realizing exactly nine distinct sup-norm nearest-neighbour distances. This file
formalizes that example, giving g_∞(3) ≥ 9; together with the repo's ≤ 9 it shows
g_∞(3) = 9 — the bound is attained.
The witness (Haynes–Ramirez, §3) #
L = ℤ³, α = (−157/10000, −742/3125, −23/400) = (−785, −11872, −2875)/50000, N = 73. The scaled
sup-norm defects M_m = max_k |r(a_k m)| (a = (−785,−11872,−2875), balanced residues mod 50000)
have eight records in the doubling window (⌈73/2⌉, 73] = (37, 73] at m = 50,51,54,55,67,68,71,72,
giving the nine distinct distances (over 50000)
11500 > 10750 > 9965 > 8912 > 8125 > 7375 > 7296 > 7088 > 7000, realized at
q = 24,23,22,19,18,6,5,2,1.
Same dynamics-free, integer-exact route as LinftyRecords.sharp_attained (d = 2), extended to
three
coordinates; everything is exact modular arithmetic — no square roots.
The L∞ rational witness α* = (−785, −11872, −2875)/50000 on 𝕋³ (Haynes–Ramirez).
Instances For
The witness defect is coordinatewise-exact: delta α* d = M/50000 with
M = max(|r₀|,|r₁|,|r₂|), the balanced residues of −785d, −11872d, −2875d mod 50000.
The witness defects delta α* m = M_m/50000 for m = 1,…,72.
Nine strictly-decreasing sample values force ≥ 9 distinct images.
nnDist as a prefix-minimum of the L∞ defect (d = 3, N = 73).
card ≥ 9 from the eight record drops at m = 50,51,54,55,67,68,71,72, realized at
q = 24,23,22,19,18,6,5,2,1 (cutoffs D = 49,50,51,54,55,67,68,71,72).
Rational → irrational upgrade by Lipschitz-openness #
The L∞ three-torus bound g_∞ ≤ 2^d+1 is sharp for d = 3. There is a Kronecker vector
α : Fin 3 → ℝ with an irrational coordinate and N ≥ 2 whose orbit on 𝕋³ realizes at least nine
distinct sup-norm nearest-neighbour distances; with SimDirichlet.nnDist_count_unconditional (≤ 9)
this gives g_∞(3) = 9 — the Shutov bound 2^d+1 is attained (Haynes–Ramirez, d ≤ 3). The
witness
is α = (−785/50000 + s, −11872/50000, −2875/50000) for a small irrational s; card ≥ 9
transports
from the rational α* by the dynamics-free Lipschitz argument.