The sum of digits of squares in \(\mathbb Z[i]\)
Let \(q\) and \(n\) be natural numbers and denote by \(s_q(n)\) the sum of digits of \(n\) written in base \(q\). Answering a question posed by \textit{A. O. Gelfond, C. Mauduit} and \textit{J. Rivat} [''La somme des chiffres des carrés,'' Acta Math. 203, No. 1, 107--148 (2009; Zbl 1278.11076)] showed that the sum \(s_q(n^2)\) of digits of squares is equidistributed in arithmetic progressions and the sequence \((\alpha s_q(n^2))_{n\in \mathbb{N}}\) is uniformly distributed modulo 1 if \(\alpha\) is irrational. This result hinges on estimates for exponential sums of the form \[ \sum\limits_{n\leq N} e(\alpha s_q(n^2)). \] The paper under review studies the analog of this question for the ring of Gaussian integers \(\mathbb{Z}[i]\). An estimate is established for exponential sums of the form \[ \sum\limits_{z\in \mathcal{D}_N} e(\alpha s_q(z^2)), \] where \(s_q(x)\) is an analog of the sum of digits function for \(\mathbb{Z}[i]\) and \((\mathcal{D}_N)_{N\in \mathbb{N}}\) is a \(\kappa\)-\(\mathbb{Z}[i]\)-sequence of subsets of \(\mathbb{Z}[i]\) with \(0<\kappa\leq 1/2\) (we refrain from giving the definition of such sequences here). Using this, the author deduces an equidistribution result on the sequence \(s_q(z^2)\) in arithmetic progressions and shows that \((\alpha s_q(z^2))_{z\in \mathbb{Z}[i]}\) is uniformly distributed modulo 1 if \(\alpha\) is irrational. The method is based on that of Mauduit and Rivat for the real case. However, a great deal of work is required to make this method working for \(\mathbb{Z}[i]\). In particular, the author needs to prove a suitable van der Corput type bound in this context.
scientific article; zbMATH DE number 5757993