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Additive Patterns in Multiplicative Subgroups

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Abstract

The study of sum and product problems in finite fields motivates the investigation of additive structures in multiplicative subgroups of such fields. A simple known fact is that any multiplicative subgroup of size at least q 3/4 in the finite field F q must contain an additive relation x + y = z. Our main result is that there are infinitely many examples of sum-free multiplicative subgroups of size Ω(p 1/3) in prime fields F p . More complicated additive relations are studied as well. One representative result is the fact that the elements of any multiplicative subgroup H of size at least q 3/4+o(1) of F q can be arranged in a cyclic permutation so that the sum of any pair of consecutive elements in the permutation belongs to H. The proofs combine combinatorial techniques based on the spectral properties of Cayley sum-graphs with tools from algebraic and analytic number theory.

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Authors and Affiliations

  1. Sackler School of Mathematics and Blavatnik School of Computer Science, Tel Aviv University, Tel Aviv, 69978, Israel

    Noga Alon

  2. Institute for Advanced Study, Princeton, NJ, 08540, USA

    Noga Alon & Jean Bourgain

Authors
  1. Noga Alon
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  2. Jean Bourgain
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Correspondence to Noga Alon.

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Research supported in part by an ERC Advanced grant, by a USA-Israeli BSF grant, by an ISF grant, by the Israeli I-Core program and by the Simonyi Fund.

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Alon, N., Bourgain, J. Additive Patterns in Multiplicative Subgroups. Geom. Funct. Anal. 24, 721–739 (2014). https://doi.org/10.1007/s00039-014-0270-y

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