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Garaev's Inequality in finite fields not of prime order

Nets Hawk Katz·Chun-Yen Shen
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Abstract

We prove a version of Garaev's sum product theorem in the set of finite fields with non-prime order. Because of the presence of subfields, this seems to require some hypotheses on the set. We work under a condition analogous to having Hausdorff dimension less than 1/2. Under these conditions, we obtain a sum-product theorem with exponent 49/48.

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Nets Hawk Katz, Chun-Yen Shen. 2007-03-22. Garaev's Inequality in finite fields not of prime order. https://arxiv.org/abs/math/0703676

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