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Statement

Notation as on the Theorem 1 page: $p$ prime, $1\le j\le J$, subintervals $I_1^{(j)},I_2^{(j)}\subseteq(0,p)$ of lengths $H$ and $K$, the $I_1^{(j)}$ pairwise disjoint.

Corollary (p. 2, unnumbered). Suppose $J\gg p^{1/3}$. Then some $j\in\{1,\dots,J\}$ has integers $x\in I_1^{(j)}$, $y\in I_2^{(j)}$ with $xy\equiv1\pmod p$, provided $H>p^{2/3}$ and $K>p^{2/3}(\log p)^2$.

The paper presents it (p. 2) as what a larger $J$ buys: it comes closer to what Hooley's conjectured bound $S(n,H)\ll H^{1/2}q^{\varepsilon}$ for incomplete Kloosterman sums (p. 1; the print writes $q$ there for the modulus) would give for a single pair, namely both lengths $\gg p^{2/3+\varepsilon}$.

A remark on the constants, of this page, not of the paper. The print gives no proof beyond placing the corollary after Theorem 1, and names no constants. Disjoint subintervals of $(0,p)$ of length $H>p^{2/3}$ number fewer than $p^{1/3}$, so the hypothesis $J\gg p^{1/3}$ can hold only with an implied constant below $1$. Substituting the thresholds in Theorem 1 needs $J\ge C\,p^{1/3}$ with Theorem 1's constant $C$. The corollary therefore follows from Theorem 1 when the thresholds on $H$ and $K$ carry suitable constant factors (for instance $J\ge c\,p^{1/3}$, $H>p^{2/3}$ and $K>(C/c)^{1/2}p^{2/3}(\log p)^2$), not with the literal thresholds for every implied constant.

Source. T. D. Browning and A. Haynes, Incomplete Kloosterman sums and multiplicative inverses in short intervals, Int. J. Number Theory 9 (2013), 481–486; read in the arXiv version 1204.6374v1, the Corollary on p. 2, Hooley's conjecture on p. 1. The edition is identified on the source card.

Read depth. Claims checked: the statement was read clause by clause against the print, and its deduction from Theorem 1 was checked here as recorded above.

Proof pointer

No proof is printed. Put $H$ and $K$ at their thresholds in the condition of Theorem 1: $p^3\log^4p/(H^2K^2)<p^3\log^4p/(p^{4/3}\cdot p^{4/3}\log^4p)=p^{1/3}$.

Dependencies

Theorem 1 of the same paper.

Bears on

No Erdős problem directly: it concerns many pairs of intervals, while Problem 445 asks about a single interval, for which the problem page uses the $J=1$ case of Theorem 1.