Resolution of T. Ward's Question and the Israel–Finch Conjecture: Precise Analysis of an Integer Sequence Arising in Dynamics. (2nd October 2014)

• Record Type: Journal Article
• Authors: GAITHER, JEFFREY; LOUCHARD, GUY; WAGNER, STEPHAN; WARD, MARK DANIEL
• Editors: Broutin, Nicolas; Fill, James Allen; Nebel, Markus; Ward, Mark Daniel
• Is Part Of: Combinatorics, probability and computing. Volume 24:Number 1(2015:Jan.)
• Page Start: 195
• Page End: 215
• Publication Date: 2014-10-02
• Subjects: Primary 05A16, -- Secondary 41A60, -- 05A17; Combinatorial analysis -- Periodicals; Probabilities -- Periodicals; Computer science -- Mathematics -- Periodicals; 511.6
• DOI: 10.1017/S0963548314000455
• Languages: English
• ISSNs: 0963-5483

Abstract

Abstract : We analyse the first-order asymptotic growth of \[ a_{n}=\int_{0}^{1}\prod_{j=1}^{n}4\sin^{2}(\pi jx)\, dx. \] The integer an appears as the main term in a weighted average of the number of orbits in a particular quasihyperbolic automorphism of a 2 n -torus, which has applications to ergodic and analytic number theory. The combinatorial structure of an is also of interest, as the 'signed' number of ways in which 0 can be represented as the sum of ϵ j j for − n ≤ j ≤ n (with j ≠ 0), with ϵ 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.