An arithmetical property of Rademacher sums. (Q1890419)

Let \(\{S_n, n \geq 1 \}\) denote the sequence of partial sums of a Rademacher sequence, and let \(N_k\) denote the \(k\)th even integer such that \(N_k \mid S_{N_{k}^{2}}\). The author shows that the sequence \(\{N_k\}\) has an exponential growth. In particular, he proves that for \(\tau > 7/8\), \(\log N_k = k/s + O(k^{\tau})\) almost surely, where \(s = 2 \sum_{j \in\mathbf Z} \exp (- 2 \pi^{2} j^{2})\). The result also extends to Bernoulli sequences.