Diophantine approximation and convergence of alternating series

Abstract:

Let $f$ be a continuous periodic function of period $2\pi$ and \[ \sum {(|{a_n}| + |{b_n}|)\log n < \infty } ,\] where ${a_n},{b_n}$ are the Fourier coefficients of $f$. Let $E$ be the set of all points $\theta$ in $[0,2\pi )$ for which the series \[ \sum {\frac {{{{( - 1)}^n}}}{n}f(n\theta )} \] does not converge. It is established here that hausdorff outer measure ${h_\alpha }(E) = 0$ for every $\alpha > 0$.