Resolution of T. Ward's Question and the Israel–Finch Conjecture: Precise Analysis of an Integer Sequence Arising in Dynamics

Abstract

We analyze the first-order asymptotic growth of a n = 1 0 n j=1 4 sin 2 (πjx)dx. The integer a n appears as the main term in a weighted average of the number of orbits in a particular quasihyperbolic automorphism of a 2n-torus, which has applications to ergodic and analytic number theory. The combinatorial structure of a n is also of interest, as the "signed" number of ways in which 0 can be represented as the sum of j j for −n ≤ j ≤ n (with j = 0), with j ∈ {0, 1}. Our result answers a question of Thomas Ward (no relation to the fourth author) and confirms a conjecture of Robert Israel and Steven Finch.Dedicated to the memory of Philippe Flajolet.

Erdos--Suranyi sequences and trigonometric integrals

“…, n} into two subsets of equal cardinality and sum (note that representations of zero of the form 0 = ±1 ± 2 ± · · · ± n correspond exactly to partitions of this type, where however the cardinalities are not necessarily equal). The asymptotic behaviour of an integral similar to the one in (1) (but with sines rather than cosines) was studied recently in [13].…”