Sum–product estimates for rational functions

By Boris Bukh, Jacob Tsimerman. 2010

49 citations recorded by the source corpus.

We establish several sum–product estimates over finite fields that involve polynomials and rational functions. First, |f(A)+f(A)+|AA| is substantially larger than |A| for an arbitrary polynomial f over 𝔽p. Second, a characterization is given for the rational functions f and g for which |f(A)+f(A)+|g(A, A)| can be as small as |A| for large |A|. Third, we show that under mild conditions on f, |f(A, A)| is substantially larger than |A|, provided |A| is large. We also present a conjecture on what the general sum–product result should be.

Subfields: Algebra and Number Theory, Artificial Intelligence, Discrete Mathematics and Combinatorics

View Paper: https://www.semanticscholar.org/paper/e1cb3e99b1f277a70dadec4d5f0f7cc5bb45a9ca

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In Algebra and Number Theory

In Artificial Intelligence

In Discrete Mathematics and Combinatorics

Citations in the corpus

Growth Estimates in Positive Characteristic via Collisions (2015, background): “They yield a considerable improvement to the state of the art, that arose from a series of questions and estimates by Bourgain, Katz, and Tao [6] and further work, e.g., by Bourgain [4] and Bukh and Tsimerman [7], see also the references contained therein.”
On the concentration of points of polynomial maps and applications (2012, methodology): “We now obtain an upper bound on M(I,J ) using the approach of [4] combined with recent results from additive combinatorics due to Bukh and Tsimerman [3].”