Sum-product Phenomenon Via Dimension

Summary · qwen2.5:32b

The study demonstrates a sum-product phenomenon in fields with abstract dimension theories, generalizing dimensions from geometric theories and Hrushovski's coarse pseudo-finite dimensions, showing that non-expansion in both sumset and product set dimensions of type-definable sets implies the existence of an equivalent definable field. The proof relies on dimensional versions of key inequalities like Ruzsa triangle and Plünnecke-Ruzsa inequalities.

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Excerpt

arXiv:2607.03384v1 Announce Type: new 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.