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

Published online by Cambridge University Press:  02 October 2014

JEFFREY GAITHER ,
GUY LOUCHARD ,
STEPHAN WAGNER  and
MARK DANIEL WARD
JEFFREY GAITHER
Affiliation:
Department of Mathematics, Purdue University, 150 North University Street, West Lafayette, IN 47907–2067, USA (e-mail: gaither.16@mbi.osu.edu)
GUY LOUCHARD
Affiliation:
Département d'Informatique, Université Libre de Bruxelles, CP 212, Boulevard du Triomphe, B-1050, Bruxelles, Belgium (e-mail: louchard@ulb.ac.be)
STEPHAN WAGNER
Affiliation:
Department of Mathematical Sciences, Stellenbosch University, Private Bag X1, Matieland 7602, South Africa (e-mail: swagner@sun.ac.za)
MARK DANIEL WARD
Affiliation:
Department of Statistics, Purdue University, 150 North University Street, West Lafayette, IN 47907–2067, USA (e-mail: mdw@purdue.edu)
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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 2n-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 ϵjj for −n ≤ j ≤ n (with j ≠ 0), with ϵj ∈ {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.

Keywords

Primary 05A16Secondary 41A6005A17
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 24 , Special Issue 1: Honouring the Memory of Philippe Flajolet - Part 3 , January 2015 , pp. 195 - 215
Copyright
Copyright © Cambridge University Press 2014 

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