On Linnik's conjecture: sums of squares and microsquares (Q2929658)

A classical theorem of C.F. Gauss states that every natural number \(n\) not of the form \(4^k(8m + 7)\) is the sum of three squares. This result inspired \textit{Yu. V. Linnik} to conjecture that when a large \(n\) is the sum of three squares, it must, in fact, have such a representation in which one of the three squares is very small [Ergodic properties of algebraic fields. Translated from the Russian by M. S. Keane. York: Springer-Verlag (1968; Zbl 0162.06801)]. In the paper under review, the author proves that a sharp version of Linnik's conjecture is true for almost all integers \(n\). Let \(E(X; Y)\) denote the number of integers \(n \in (X/2, X]\), with \(n \equiv 1, \pm 2, \pm 3 \pmod 8\), that cannot be represented as \(n = x_1^2 + x_2^2 + x_3^2\), with \(x_1,x_2,x_3\) positive integers and \(x_3 \leq Y\). The author proves that if \((\log X)(\log\log X)^2 \leq Y \leq (\log X)^{1+\delta}\) (with \(\delta > 0\) fixed), then one has \(E(X; Y) \ll XY^{-1}(\log X)(\log\log X)^2\). He also obtains a similar result on sums of four squares, two of which are small. The proofs of the results use the Hardy-Littlewood circle method and ideas for dealing with ``slim exceptional sets'' from the author's earlier work [Proc. Lond. Math. Soc. (3) 85, No. 1, 1--21 (2002; Zbl 1039.11066)].