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arXiv · 2607.03384

Sum-product Phenomenon Via Dimension

Yuyan He·Sergei Starchenko
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Abstract

We show a sum-product phenomenon in fields equipped with abstract dimension theories, which simultaneously generalizes the dimensions in geometric theories and Hrushovski's coarse pseudo-finite dimensions. More precisely, we show that for type-definable sets of positive non-zero dimension, non-expansion in dimension of both the sumset and product set implies the existence of a definable field in the same dimension. Main ingredients of the proof include dimensional analogues of the Ruzsa triangle inequality and the Pl\"unnecke-Ruzsa inequality.

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Yuyan He, Sergei Starchenko. 2026-07-03. Sum-product Phenomenon Via Dimension. https://arxiv.org/abs/2607.03384

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