On extreme values for the Sudler product of quadratic irrationals

Abstract

Given a real number α and a natural number N , the Sudler product is defined by P N (α) = N r=1 2 sin(π (rα)) . Denoting by F n the n-th Fibonacci number and by φ the Golden Ratio, we show that for F n−1 N < F n , we have P Fn−1 (φ) P N (φ) P Fn−1 (φ) and min N 1 P N (φ) = P 1 (φ), thereby proving a conjecture of Grepstad, Kaltenböck and Neumüller. Furthermore, we find closed expressions for lim inf N →∞ P N (φ) and lim sup N →∞ P N (φ) N whose numerical values can be approximated arbitrarily well. We generalize these results to the case of quadratic irrationals β with continued fraction expansion β = [0; b, b, b . . .] where 1 b 5, completing the calculation of lim inf N →∞ P N (β), lim sup N →∞ P N (β) Nfor β being an arbitrary quadratic irrational with continued fraction expansion of period length 1.

Metric density results for the value distribution of Sudler products

“…This approach was first used in the special case for α being the Golden Ratio in [13] and made more explicit and general in subsequent works in this area, e.g. [3,4,14,15,16]. Defining…”