Diophantine Approximations and the Convergence of Certain Series

2015/06/18 by Begunts, Alexander, Goryashin, Dmitry
#11J82 #40A05 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1506.05746

Abstract

Consider two series ∑_n=1^∞(sinⁿπθn)/(n^α), ∑_n=1^∞(cosⁿπθn)/(n^α). We show that number-theoretical properties of θ have a strong effect on the convergence when 0\frac12 then both series converge absolutely for almost all real θ. Finally, we construct such an everywhere dense set of θ that both series diverge when α≤ 1.

Related

• Sums, series, and products in Diophantine approximation
• Convergence of a sinusoidal infinite series from Borwein, Bailey, and Girgensohn
• On convergence of the Flint Hills series
• Uniform convergence criterion for non-harmonic sine series
• Bounds on Irrationality Measures and the Flint-Hills Series