Moment Estimates and Discrepancy for Sums of Square Roots Modulo One

Abstract

Let $k\ge 2$ be fixed. We study the distribution modulo one of the $n^k$ sums \begin{equation*} \sqrt{a_1} + \cdots + \sqrt{a_k}, \qquad 1\le a_1, \dots, a_k \le n, \end{equation*} counted with multiplicity. For \begin{equation*} S(h,n) = \sum_{n/2\le a\le n} \mathbf{e}(h\sqrt{a}), \qquad \mathbf{e}(x) = \exp(2\pi i x), \end{equation*} we prove second- and fourth-moment estimates matching the diagonal scale up to a factor $n^\varepsilon$. More precisely, \begin{equation*} \sum_{H/2\le h\le H} \left| S(h,n) \right|^2 \ll_{\varepsilon,\delta} Hn^{1+\varepsilon} \end{equation*} uniformly for $H\ge n^{1/2+\delta}$, and \begin{equation*} \sum_{H/2\le h\le H} \left| S(h,n) \right|^4 \ll_{\varepsilon,\delta} Hn^{2+\varepsilon} \end{equation*} uniformly for $n^{1/2+\delta} \le H \le n^{2/3}$, where $0<\delta<1/6$ in the fourth-moment estimate. Combining the second-moment bound with pointwise exponential-sum estimates and the Erd\H{o}s--Tur\'an inequality, we obtain \begin{equation*} D_k(n) \le n^{-\rho_k+o(1)}, \qquad \rho_k = \frac{71k+26}{26k+116}, \end{equation*} as $n\to\infty$, where $D_k(n)$ denotes the discrepancy with respect to arbitrary subintervals of $[0,1)$.