Shparlinski 2012 modular hyperbolas

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theorem_13: Shparlinski's uniform-distribution count for the modular hyperbola xy = a mod m, for every modulus m and every a coprime to m, with x in an interval of length X and y in an interval of length Y that may depend on x; the error term O(m^{1/2+o(1)}) does not depend on X, Y or the intervals.

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Igor E. Shparlinski, Modular Hyperbolas. Japanese Journal of Mathematics 7 (2012), 235-294. doi:10.1007/s11537-012-1140-8. arXiv:1103.2879. The copy read for this card is arXiv:1103.2879v4 (3 June 2012); the theorem, formula and page numbers on this card refer to it. The arXiv record names arXiv's non-exclusive distribution license (arXiv:1103.2879), every other right reserved.

This survey collects results about the distribution and geometric properties of the point set H_{a,m} of solutions to xy congruent to a mod m, with gcd(a,m) = 1, together with multivariate generalizations, applications, and a number of open problems of varying difficulty. It explains how many results proved case by case in the literature follow as simple corollaries of one general uniformity-of-distribution statement, Theorem 13, which counts points of H_{a,m} with x in an interval X of length X and y in per-x intervals of length Y and is derived directly from bounds on Kloosterman sums; specializing to a box (the same y-interval for every x) gives formula (10), #H_{a,m}(X, Y) = (phi(m)/m^2) XY + O(m^{1/2+o(1)}). Its stated main purpose, however, is to outline the subtler arguments that use special properties of the congruence xy congruent to a mod m and do not extend to other congruences. Proofs are sketched rather than given in full, and error terms typically carry m^{o(1)} factors.

Source: https://arxiv.org/abs/1103.2879.

Bears on. #158: no result of the paper bears on the problem's question.

Results. Labels and pages are those of arXiv:1103.2879v4. Read status: claims checked for the page below; the proof was read but not checked step by step.

• Theorem 13 (p. 13; proof pp. 13--14): for an interval X = {U+1, ..., U+X} and, for each x in it, an interval Y_x = {V_x+1, ..., V_x+Y}, where m > X >= 1, m > Y >= 1 and U, V_x >= 0 are integers, and for every integer m >= 1 and every a with gcd(a,m) = 1, the number of points (x,y) of H_{a,m} with x in X and y in Y_x is (phi(m)/m^2) XY + O(m^{1/2+o(1)}), derived from the Kloosterman-sum bound (1) (p. 3). The same page records the box case, formula (10) (p. 14), and the survey's remark (p. 14) that making Theorem 13 or (10) nontrivial for XY < m^alpha with some fixed alpha < 3/2 seems out of reach at present.

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