Critical convergence and Hausdorff measures for generalized Flint Hills series

Abstract

We study the generalized Flint Hills series ${\mathcal{F}}_{s,t}(x)={\sum }_{n\ge 1}{n}^{-s}|\sin (\pi nx){|}^{-t}$ for $s>0$ and $t>1$. An explicit comparison with a series over continued-fraction denominators yields the Hausdorff dimension $\min \{1,2t/(s+t)\}$ of its divergence set. At each critical exponent $\tau =1+s/t>2$ we construct numbers of irrationality exponent $\tau$ realizing both convergence and divergence; the convergent examples establish Meiburg's conjecture in the range $t>1$. Both parts of the critical fibre have Hausdorff dimension $2/\tau$. For $s>t$ and $h_{\kappa }(r)=r^{2/\tau }(\log (1/r){)}^{\kappa }$ we prove that the divergence set has zero $h_{\kappa }$-measure for $\kappa <-1$ and infinite measure for $\kappa \ge -1$. The divergent part of the critical fibre satisfies the same law, whereas its convergent part has infinite measure for every $\kappa$. The key estimate selects a rapidly growing subsequence of convergent denominators and gives a double-logarithmic bound on the approximation error. The convergence of the classical Flint Hills series ${\sum }_{n\ge 1}(n^3{\sin }^{2}n{)}^{-1}$ remains undecided.

18 pages