Sums of square roots that are close to an integer (Q6556213)

In this article, the author examines the following question: Given \(k,n \in \mathbb{N}\), and suppose that integers \(1 \leq a_1,\ldots,a_k \leq n\) are given with the property that \(\sum_{i = 1}^k \sqrt{a_i} \notin \mathbb{N}\), how close to an integer could such a sum come? While surveying results on similar-looking problems such as minimizing the distances\N\[\N\left \vert\sum_{i = 1}^k \sqrt{a_i} - \sum_{i = 1}^k \sqrt{b_i} \right\rvert,\N\]\Nwhich appears in numerical analysis, or considering the gap and fine-scale distributions of \(\sqrt{n} \pmod 1\) that has a rich body of literature in dynamics and number theory, the author proves the following main result:\N\NThere exists \(c > 0\) such that for any \(k \in \mathbb{N}\) fixed, there exists \(c_k > 0\) such that for all \(n \in \mathbb{N}\), there exist \(1 \leq a_1, \ldots, a_k \leq n\) with \N\[\N0 < \lVert \sqrt{a_1} + \ldots + \sqrt{a_k} \rVert \leq c_k n^{-c \cdot k^{1/3}}.\tag{1}\N\]\NActually, the author proves the stronger result that the largest gap arising from the set \(\left\{\sqrt{a_1} + \ldots + \sqrt{a_k} \pmod 1: 1 \leq a_1, \ldots, a_k \leq n\right\}\) is bounded by the quantity described above. The proof of (1) uses standard tools from the theory of exponential sums (i.e. van der Corput and Vinogradov method).\N\NAs to be expected, the bottleneck for improvements in the author's approach is getting a better control on the exponential sums. While some vaguely formulated form of exponent pair hypothesis could provide \(n^{-k^{1/2 + \varepsilon}}\) and modeling of \(\sqrt{a_i} \pmod 1\) with uniformly distributed random variables suggests a bound to be of the form \(\approx n^{-k}\), such a result would still be far away from matching the lower bound: By an elementary approach presented in the article, the best lower bound known is of the form \(\gg n^{1/2 - 2^{k-1}}\), and the author claims that a polynomial bound in \(k\) seems out of reach.