A problem of Orlicz in the Scottish Book (Q1819304)

The problem mentioned in the title is to give an explicit example of a trigonometric series \(\sum (a_ n\cos (nx)+b_ n\sin (nx))\) that diverges everywhere, but for which \[ \sum^{\infty}_{n=1}(| a_ n|^{2+\epsilon}+| b_ n|^{2+\epsilon}) \] converges for every \(\epsilon >0\). As the authors note, the existence of such series is known, but no explicit construction had been given so far. In the paper under review the authors show that the series defined by \[ a_ n=\frac{\cos (2\pi \alpha n\quad \log n)}{\sqrt{n}},\quad b_ n=- \frac{\sin (2\pi \alpha n\quad \log n)}{\sqrt{n}} \] has the above properties for certain values of \(\alpha\) such as \(\alpha =1/\log 2\). The proof depends on the following exponential sum estimate: For N sufficiently large and real numbers \(\alpha >0\) and \(\beta\) the inequality \[ | \sum_{e^{-1/\alpha}N<n\leq N}\exp (2\pi i\alpha n\quad \log n+\beta n)| \geq C_{\alpha} \sqrt{N} \] holds with a positive constant \(C_{\alpha}\) depending only on \(\alpha\). The authors prove this estimate using van der Corput's method of transforming the exponential sum first into an integral and then evaluating this integral by the stationary phase method.