On additive shifts of multiplicative subgroups

By Shkredov I.D., Vyugin I.V. 2011

60 citations recorded by the source corpus.

Generalizing a result of S.V. Konyagin and D.R. Heath–Brown, we prove, in particular, that for any multiplicative subgroup R ⊆ Z/pZ and any nonzero elements μ1, . . . , μk the following holds |R(R + μ1) ⋂ · · ·(R+ μk)| ≪k |R| 1 2 +αk , provided by 1 ≪k |R| ≪k p1−βk , where αk, βk are some sequences of positive reals and αk, βk → 0, k → ∞. Besides we show that for an arbitrary subgroup R, |R| ≪ p1/2 one have |R±R| ≫ |R|5/3 log |R|.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics

View Paper: https://www.semanticscholar.org/paper/68891c43c99e59dd081eaed9cd98e86fb321091e

How researchers see this paper

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In Algebra and Number Theory

1. Ilya D. Shkredov
2. Oliver Roche-Newton
3. M. Rudnev
4. Tom Sanders
5. Ben Joseph Green

In Discrete Mathematics and Combinatorics

1. Ilya D. Shkredov
2. M. Rudnev
3. Oliver Roche-Newton
4. Sophie Stevens
5. Frank de Zeeuw
6. Brendan Murphy
7. Tom Sanders
8. Ben Joseph Green

Citations in the corpus

• NEW RESULTS ON SUM‐PRODUCT TYPE GROWTH OVER FIELDS (2017, unknown): “for A ⊂ R satisfying |A/A| ≪ |A|, and the fifth listed author’s and Vyugin’s lower bound [47] for small multiplicative subgroups Γ of Fp: |Γ− Γ| & |Γ|, which improves the lower bound of |Γ− Γ| ≫ |A|3/2 established by Heath-Brown and Konyagin [15].”
• New sum-product type estimates over finite fields (2014, methodology): “For genuine multiplicative subgroups in prime fields this was done in [26]; for approximate ones over R and C – in [15].”