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The Algebraicity of Ill-Distributed Sets

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Abstract

We show that every set \({S \subseteq [N]^d}\) occupying \({\ll p^{\kappa}}\) residue classes for some real number \({0 \leq \kappa < d}\) and every prime p, must essentially lie in the solution set of a polynomial equation of degree \({\ll ({\rm log} N)^C}\), for some constant C depending only on \({\kappa}\) and d. This provides the first structural result for arbitrary \({\kappa < d}\) and S.

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Authors and Affiliations

  1. Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, 1428, Buenos Aires, Argentina

    Miguel N. Walsh

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  1. Miguel N. Walsh
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Correspondence to Miguel N. Walsh.

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M. N. Walsh was partially supported by a CONICET doctoral fellowship.

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Walsh, M.N. The Algebraicity of Ill-Distributed Sets. Geom. Funct. Anal. 24, 959–967 (2014). https://doi.org/10.1007/s00039-014-0286-3

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  • DOI: https://doi.org/10.1007/s00039-014-0286-3

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