Title: Additive energy and a large sieve inequality for sparse sequences
Abstract: MathematikaVolume 68, Issue 2 p. 362-399 RESEARCH ARTICLE Additive energy and a large sieve inequality for sparse sequences Roger C. Baker, Roger C. Baker Department of Mathematics, Brigham Young University, Provo, Utah, USASearch for more papers by this authorMarc Munsch, Corresponding Author Marc Munsch [email protected] DIMA, Università degli Studi di Genova, Genova, Italy Correspondence Marc Munsch, DIMA, Università degli Studi di Genova, via Dodecaneso 35, Genova I-16146, Italy. Email: [email protected]Search for more papers by this authorIgor E. Shparlinski, Igor E. Shparlinski Department of Pure Mathematics, University of New South Wales, UNSW, Sydney, NSW, AustraliaSearch for more papers by this author Roger C. Baker, Roger C. Baker Department of Mathematics, Brigham Young University, Provo, Utah, USASearch for more papers by this authorMarc Munsch, Corresponding Author Marc Munsch [email protected] DIMA, Università degli Studi di Genova, Genova, Italy Correspondence Marc Munsch, DIMA, Università degli Studi di Genova, via Dodecaneso 35, Genova I-16146, Italy. Email: [email protected]Search for more papers by this authorIgor E. Shparlinski, Igor E. Shparlinski Department of Pure Mathematics, University of New South Wales, UNSW, Sydney, NSW, AustraliaSearch for more papers by this author First published: 12 April 2022 https://doi.org/10.1112/mtk.12140Read the full textAboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Abstract We consider the large sieve inequality for sparse sequences of moduli and give a general result depending on the additive energy (both symmetric and asymmetric) of the sequence of moduli. For example, in the case of monomials f ( X ) = X k $f(X) = X^k$ this allows us to improve, in some ranges of the parameters, the previous bounds of S. Baier and L. Zhao (2005), K. Halupczok (2012, 2015, 2018) and M. Munsch (2020). We also consider moduli defined by polynomials f ( X ) ∈ Z [ X ] $f(X) \in \mathbb {Z}[X]$ , Piatetski–Shapiro sequences and general convex sequences. We then apply our results to obtain a version of the Bombieri–Vinogradov theorem with Piatetski–Shapiro moduli improving the level of distribution of R. C. Baker (2014). REFERENCES 1Y. Akbal, Rough values of Piatetski-Shapiro sequences, Monatsh. Math. 185 (2018), 1– 15. 2S. Baier and L. Zhao, Large sieve inequality with characters for powerful moduli, Int. J. Number Theory 1 (2005), 265– 279. 3S. Baier and L. Zhao, Bombieri-Vinogradov type theorems for sparse sets of moduli, Acta Arith. 125 (2006), 187– 201. 4S. Baier and L. Zhao, An improvement for the large sieve for square moduli, J. 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