Browning and Haynes (2013): Incomplete Kloosterman sums

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corollary: The unnumbered corollary of Browning and Haynes's Theorem 1: with J >> p^{1/3} pairs of intervals, the first ones disjoint, some pair holds an inverse pair mod p once H > p^{2/3} and K > p^{2/3}(log p)^2, closer to what Hooley's conjectured bound would give.

theorem_1: Browning and Haynes's main theorem: for J pairs of subintervals of (0,p) of lengths H and K, the first intervals pairwise disjoint, some pair holds x, y with xy = 1 mod p once J >> p^3 log^4 p/(H^2K^2); the case J = 1 is the two-interval criterion HK >> p^{3/2} log^2 p.

theorem_2: Browning and Haynes's mean value theorem: over disjoint subintervals of (0,p) of lengths in (H/2, H], the squares of the incomplete Kloosterman sums of the inverses sum to at most 2^12 p log^2 H, for every nonzero residue l; the input to Theorem 1.

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T. D. Browning and A. Haynes, Incomplete Kloosterman sums and multiplicative inverses in short intervals, International Journal of Number Theory 9 (2013), 481–486, DOI 10.1142/S1793042112501448.

The paper asks when integers $x,y$ in prescribed subintervals of $(0,p)$, $p$ prime, satisfy $xy\equiv1\pmod p$. Heuristically lengths $\gg p^{1/2}$ should suffice; the paper records (p. 1) that the best result to date, highlighted by Heath-Brown, needs $|I_1|\cdot|I_2|\gg p^{3/2}\log^2p$, and that Hooley's conjectured bound for incomplete Kloosterman sums would allow lengths $\gg p^{2/3+\varepsilon}$. Its main result, Theorem 1 (p. 2), takes $J$ pairs of intervals of lengths $H$ and $K$, the first ones pairwise disjoint, and finds an inverse pair in one of them once $J\gg p^3\log^4p/(H^2K^2)$; $J=1$ gives back the two-interval criterion. An unnumbered Corollary (p. 2) takes $J\gg p^{1/3}$, $H>p^{2/3}$ and $K>p^{2/3}(\log p)^2$. The engine is Theorem 2 (p. 2), a mean value bound $2^{12}p\log^2H$ for the squares of incomplete Kloosterman sums over disjoint intervals of comparable length, proved in Section 2 (pp. 2--5) from Weil's bound by the method of Heath-Brown's work on Burgess's bounds; Section 3 (pp. 5--6) proves Theorem 1.

The copy read for this card is arXiv:1204.6374v1, dated 28 April 2012, six pages. Browning's author publication list supplies the journal identity; the journal PDF was not compared, and the labels and pages cited are the arXiv version's.

Read status. Claims checked: Theorem 1, Theorem 2 and the Corollary (p. 2) and the $J=1$ remark were read clause by clause against the print. The proofs of Theorems 1 and 2 (pp. 2--6) were read for structure only, not verified. The Theorem 2 page records that the printed bound fails for $H$ just above $1$, outside the range $H\ge4$ that the proof treats, and the Corollary page records that the Corollary follows from Theorem 1 only with suitable constant factors in its thresholds; both are observations of those pages, not of the paper.

Bears on. #445: the $J=1$ case of Theorem 1 is the two-interval criterion from which the problem page deduces, by a short reduction stated there, an inverse pair in every interval $(n,n+p^c)$ for each fixed $c>3/4$ and all large $p$; the criterion gives nothing at $c=3/4$ or below. Theorem 2 and the Corollary bear on the problem only through Theorem 1.

Results. Theorem 1, Theorem 2 and the Corollary.

Read artifact. The arXiv record names arXiv's non-exclusive distribution license (arXiv:1204.6374), every other right reserved.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.