Critical convergence and Hausdorff measures for generalized Flint Hills series

Source abstract

We study the generalized Flint Hills series $\mathcal{F}_{s,t}(x)=\sum_{n\ge1} n^{-s}|\sin(πnx)|^{-t}$ 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 $τ=1+s/t>2$ we construct numbers of irrationality exponent $τ$ 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 $2/τ$. For $s>t$ and $h_κ(r)=r^{2/τ}(\log(1/r))^κ$ we prove that the divergence set has zero $h_κ$-measure for $κ<-1$ and infinite measure for $κ\ge-1$. The divergent part of the critical fibre satisfies the same law, whereas its convergent part has infinite measure for every $κ$. 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 $\sum_{n\ge1}(n^3\sin^2 n)^{-1}$ remains undecided.