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Statement

Let $p$ be a prime (p. 1). For $n$ prime to $p$, $\overline n$ denotes the inverse of $n$ modulo $p$, and $e(x)=e^{2\pi ix}$ (the print uses both without defining them).

Theorem 2 (p. 2). If $I_1,\dots,I_J\subseteq(0,p)$ are disjoint subintervals with $H/2<|I_j|\le H$ for each $j$, then for every $\ell\in(\mathbb Z/p\mathbb Z)^*$,

$$\sum_{j=1}^{J}\Biggl|\sum_{n\in I_j}e\Bigl(\frac{\ell\overline n}{p}\Bigr)\Biggr|^2 \le2^{12}\,p\log^2H.$$

The paper remarks (p. 2) that $J=1$ recovers, up to a constant factor, the Weil-type bound (1) of p. 1 for a single incomplete Kloosterman sum.

A boundary remark of this page, not of the paper. The proof (p. 3) assumes $H\ge4$, saying the result is trivial otherwise. For $H$ just above $1$ the printed bound fails: for $1<H<9/5$, the $J=p-1$ intervals of length $9/10$ centred at $1,\dots,p-1$ are disjoint subintervals of $(0,p)$ with $H/2<|I_j|\le H$, each inner sum has modulus $1$, so the left side is $p-1$, while $2^{12}p\log^2H$ tends to $0$ as $H\to1^+$. The proof covers $H\ge4$. In Theorem 1 the hypothesis on $J$, together with $JH\le p$ (the first intervals are disjoint) and $K\le p$, forces $H\gg\log^4p$.

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, Theorem 2 on p. 2, the proof in Section 2 on pp. 2--5. The edition is identified on the source card.

Read depth. Claims checked: the statement was read clause by clause against the print. The proof (pp. 2--5) was read for its structure only, not verified.

Proof pointer

Section 2, pp. 2--5. An unnumbered Lemma (p. 3), cited on p. 5 as "Lemma 2" [sic], bounds the complete second moment: for $H\in\mathbb N$ and $\ell\in(\mathbb Z/p\mathbb Z)^*$, $\sum_{n=1}^p|S(n,H)|^2\le H^2/p+8pH$, where $S(n,H)$ is the incomplete Kloosterman sum of $e(\ell\overline m/p)$ over $n<m\le n+H$, $m\not\equiv0\pmod p$; its proof expands the square, completes with additive characters and uses Weil's bound for the complete Kloosterman sums $K(\ell,a;p)$. The rest follows the proof of Heath-Brown's Theorem 2 for character sums (the paper's reference [3]): after spacing the intervals by taking odd and even indices separately, each interval sum is bounded by an average of maximal sums $\max_{k\le2H}|S(n,k)|$ (displays (3), (4)), and a dyadic decomposition of $k$ with Cauchy's inequality reduces these maxima to the Lemma, giving $2^8(H^2/p+2pH)\log^2H$.

Dependencies

Weil's bound for complete Kloosterman sums; the method of Heath-Brown, Burgess's bounds for character sums (the paper's reference [3], arXiv:1203.5219), Theorem 2.

Bears on

No Erdős problem directly. It is the analytic input to Theorem 1, whose $J=1$ case the page of Problem 445 applies.