Sum-product estimates for rational functions

TL;DR

This paper proves new sum-product estimates involving polynomials and rational functions over finite fields, revealing when these sets grow significantly larger than the original set, and characterizing cases of minimal growth.

Contribution

It introduces novel sum-product bounds for rational functions over finite fields and characterizes functions with minimal sum-product growth.

Findings

|f(A)+f(A)|+|AA| is much larger than |A| for any polynomial f

Characterization of rational functions f,g with minimal sum-product size

Under mild conditions, |f(A,A)| is substantially larger than |A| for large A

Abstract

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 F_p. 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.