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Statement

Theorem (stated in the introduction, p. 168, unnumbered; proved in Section 2, pp. 170-171). Let $r_1<r_2<\cdots$ be an infinite sequence of positive integers, and suppose that for infinitely many $N$ the number of terms $r_i\le N$ is greater than $N^{1-(c_4/\log\log N)}$, where $c_4<\tfrac12\log2$. Then for infinitely many $M$ the number of solutions of

$$r_j^2-r_i^2=M$$

is greater than $M^{c_5/\log\log M}$, where $c_5$ depends only on $c_4$.

The paper presents this (p. 168) as a generalization of the result proved in Section 1 of Part I, J. London Math. Soc. 12 (1937), 133-136. Among Part I's results the introduction recalls that $m=p^2-q^2$ has more than $m^{c_1/\log\log m}$ solutions in primes for infinitely many $m$.

Read depth. Claims checked: the statement was read clause by clause on the printed page. The proof was read but not checked step by step. Nothing here is independently reviewed.

Proof pointer

Pages 170-171. For large $N$ the paper takes $A=3\cdot5\cdots p_\mu$, the product of the first $\mu$ odd primes, with $3\cdots p_\mu\le N<3\cdots p_\mu p_{\mu+1}$. Squares fall into $z=\prod_{i\le\mu}\tfrac12(p_i+1)$ classes modulo $A$, and $z<A^{1-(c_8/\log\log A)}$ for every $c_8<\log2$. Cauchy-Schwarz over these classes bounds below the number $S$ of pairs $r_i<r_j\le N$ with $A\mid r_j^2-r_i^2$; choosing $c_8>2c_4$, which the hypothesis $2c_4<\log2$ allows, gives $S>\tfrac14N^{1+(c_9/\log\log N)}$ with $c_9=c_8-2c_4>0$. All these differences are positive and below $N^2$, so some multiple $M\le N^2$ of $A$ carries more than $M^{c_5/\log\log M}$ of them.

Dependencies

The prime number theorem, for $\mu>c\log A/\log\log A$ with any $c<1$; the count $\tfrac12(p+1)$ of square classes modulo an odd prime $p$.

Bears on

This page records no Erdős problem that the theorem bears on.