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Algebraic methods in sum-product phenomena

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Abstract

We classify the polynomials f(x, y) ∈ ℝ[x, y] such that, given any finite set A ⊂ ℝ, if |A + A| is small, then |f(A,A)| is large. In particular, the following bound holds: |A + A‖f(A,A)| ≳ |A|5/2. The Bezout theorem and a theorem by Y. Stein play an important role in our proof.

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  1. Department of Mathematics, Indiana University, Bloomington, IN, 47405, USA

    Chun-Yen Shen

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  1. Chun-Yen Shen
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Correspondence to Chun-Yen Shen.

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Shen, CY. Algebraic methods in sum-product phenomena. Isr. J. Math. 188, 123–130 (2012). https://doi.org/10.1007/s11856-011-0096-3

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  • DOI: https://doi.org/10.1007/s11856-011-0096-3

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