Modular hyperbolas and Beatty sequences

JOURNAL OF NUMBER THEORY

卷:208

Article

Technau, Marc¹ 

[1] Grnz Univ Technol, Inst Anal & Number Theory, Kopernikusgasse 24, A-8010 Graz, Austria

关键词: Beatty sequence;    Modular hyperbola;    Kloosterman sum;   

DOI  :  10.1016/j.jnt.2019.07.022

来源: Elsevier

【 摘 要 】

Bounds for max{m,( m) over tilde} subject to m, (m) over tilde is an element of Z boolean AND [1,p), p prime, z indivisible by p, m (m) over tilde (math) z mod p and m belonging to some fixed Beatty sequence {left perpendicularn alpha + beta right perpendicular : n is an element of N} are obtained, assuming certain conditions on alpha. The proof uses a method due to Banks and Shparlinski. As an intermediate step, bounds for the discrete periodic autocorrelation of the finite sequence 0, e(p)(y (1) over bar), e(p)(y (2) over bar), ..., e(p)(y((p - 1) over bar)) on average are obtained, where e(p)(t) = exp(2 pi it/p) and m (m ) over tilde (math) mod p. The latter is accomplished by adapting a method due to Kloosterman. (C) 2019 Elsevier Inc. All rights reserved.

【 授权许可】

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