A distribution result for slices of sums of squares (Q2777986)

The primary purpose of this paper is to establish some results concerning the distribution of solutions of the Diophantine equation \(m=x_1^2+\cdots+x_s^2\) for slices of consecutive integers \(m\), where \(s=3\) or 4. The authors announce that these results shall be applied in their forthcoming paper related to the study of the Schrödinger equation. NEWLINENEWLINENEWLINEFor \(0\leq x\leq 1\), let \(R_{2,s}(m;x)\) denote the number of solutions of the above Diophantine equation in non-negative integers \(x_1,\ldots,x_s\) satisfying \(0\leq x_1\leq x\sqrt{m}\), and write \(R_{2,s}(m)=R_{2,s}(m;1)\). Also, put \(\mu_{2,4}^{(h)}(x)=4\pi^{-1}(1-x^2)^{1/2}\) and \(\mu_{2,3}^{(h)}(x)=1\). Then, it is proved in this paper that, for \(s=3\) and 4, one has NEWLINE\[NEWLINE \sum_{m=n-h+1}^n R_{2,s}(m;x)= \Biggl( \int_0^x\mu_{2,s}^{(h)}(y) dy+o(1)\Biggr) \sum_{m=n-h+1}^n R_{2,s}(m), NEWLINE\]NEWLINE as \(n\rightarrow\infty\), provided that \(h=h(n)\geq n^{(4-s)/4+\varepsilon}\) with any fixed positive \(\varepsilon\). Namely, for \(s=3\) and 4, each normalized coordinate of solutions of the above Diophantine equation admits a limit law with density \(\mu_{2,s}^{(h)}(x)\) (with respect to the Lebesgue measure on \([0,1]\)). A similar thorem is proved for a more general situation, that is, for the number of solutions of the same Diophantine equation subject to \(y_i\sqrt{m}\leq x_i\leq z_i\sqrt{m}\) (\(1\leq i\leq s\)) in place of \(R_{2,s}(m;x)\) above, where \(y_i\) and \(z_i\) are any real numbers with \(0\leq y_i\leq z_i\leq 1\). The proofs are based on Kloosterman's method. The paper includes some comments, in relation to theorems already established in this direction such as the deep work of \textit{W. Duke} [Invent. Math. 92, 73--90 (1988; Zbl 0628.10029)]. By a reasonable heuristic argument, the authors conjecture that the restriction on \(h\) in the above main theorems of this paper would be replaced, in truth, by \(h\geq 2\) when \(s=4\), and \(h\geq 3\) when \(s=3\). NEWLINENEWLINENEWLINERelated results on sums of \(k\)th powers are also recorded. Moreover, it is pointed out that results on such limit distributions may be established for various additive problems.