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math.DS — Dynamical Systems

No arrivals: complete periodicity and degenerate orbits in rational right triangles

Contributed by Joshua Boorstein

Boshernitzan conjectured in 1992 that in a rational right or isosceles triangle every direction of an explicit finite set is purely periodic: every billiard orbit in it that avoids the vertices is periodic. For a right triangle with acute angle $P\pi/2Q$ in lowest terms the set is two classes of directions, each containing the perpendicular to a side, and the conjecture says that both are completely periodic. It fails at one known right triangle, $(4/15, 7/30, 1/2)\pi$ . We prove it, when $\min(P, Q - P) \geq 2$ , whenever $P + 2$ and $P - 2$ are both prime to $Q$ , a set of density about $0.62$ , and whenever $Q$ has at most one odd prime factor, with a computer-assisted check of finitely many cases; and, through the fold, for the corresponding isosceles triangles. Everything turns on one object, an orbit from a vertex to a vertex that never strikes a side at a right angle: if there is none, both classes are completely periodic, and at odd $P$ the retracing band of a companion paper exists. Such an orbit satisfies an identity among roots of unity whose coefficients have signs fixed by the geometry, and Galois conjugation — or, at a single odd prime, a slowest-rotation argument — makes it impossible. When neither $P + 2$ nor $P - 2$ is a unit modulo $Q$ , no argument from these inputs can succeed, and the counterexample lies there. A second method reads the poles of a minimal component as a word of a binary code attached to $Q$ ; it reaches further composite $Q$ and, with interval arithmetic, the triangles $(4, Q - 4, Q)$ at every $Q \equiv 3 \pmod{4}$ . Beyond right triangles we prove, with computer assistance, that Boshernitzan's Conjecture 2.2(a) fails for infinitely many triangles.

Author comments: First version. Companion papers: "The orphan theorem" (hexagon:2611.00191) and "The necklace of a rational right triangle" (hexagon:2611.00213), whose results this paper imports; "Non-perpendicular periodic orbits in mirror directions of right triangles" in preparation. Code: https://doi.org/10.5281/zenodo.24037744

Primarily AI-generated textHuman understanding: some partsBoshernitzan conjectureGalois conjugationcomplete periodicitycyclic codesgeneralized diagonalshalf-translation surfacesisosceles trianglesrational billiardsright trianglesvanishing sums of roots of unity

Nov 10, 202694 pages

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hexagon:2611.00246v1

math.NT — Number Theory

Second moments of pattern counts in dual Melas codewords

Contributed by David Niedbala Giraudin

Let $\alpha$ be a primitive element of $\mathbb{F}_{2^k}$ and, for $c\in\mathbb{F}_{2^k}^*$ , let $S_c=(\mathrm{Tr}(\alpha^i+c\alpha^{-i}))_{0\le i\le 2^k-2}$ be the corresponding codeword of the dual Melas code. Let $X_{k,c}(w)$ be the number of occurrences of a uniformly random pattern $w\in\mathbb{F}_2^k$ in $S_c$ . Sac Himelfarb and Schwartz conjectured that $X_{k,c_k}$ converges to the Poisson law $\mathcal P(1)$ for every choice of $c_k\in\mathbb{F}_{2^k}^*$ ; they proved that the second factorial moment satisfies $\limsup M_2\le 2$ , and $M_2\to1$ for almost every sequence $(c_k)$ . We prove that $|M_2^{(k,c)}-1|\le 2^{-(1/22+o(1))k}$ uniformly in $c\in\mathbb{F}_{2^k}^*$ and in the primitive element $\alpha$ . Thus the second factorial moment converges to $1$ for every sequence $(c_k)$ . The proof expresses $M_2$ through a norm form of the quadratic extension $\mathbb{F}_2(x)/\mathbb{F}_2(x+x^{-1})$ , which turns it into a bilinear Kloosterman sum of full length, and bounds the latter in characteristic two. We also show that the counts $N_w(S_c)$ , as functions of $(w,c)$ , are the fibre sizes of a family of quadratic maps, and deduce that for almost every sequence of patterns $w_k$ the first two factorial moments of $c\mapsto N_{w_k}(S_c)$ converge to $1$ , as they would under the Poisson law conjectured by Sac Himelfarb and Schwartz for fixed patterns.

Author comments: 7 pages. Proves, uniformly in c, the second-moment case of Conjecture 35 of Sac Himelfarb-Schwartz (arXiv:2610.38017), and the first two moments of their fixed-pattern conjecture for almost every pattern. Ancillary verifier VERIF_MELAS_M2.py (pure Python).

Primarily AI-generated textHuman understanding: no partsKloosterman sumsLFSR sequencesPoisson randomnessdual Melas codesfunction fields

Nov 10, 20267 pages

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No arrivals: complete periodicity and degenerate orbits in rational right triangles

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