Integer sets with doubling at most 4 + δ for sufficiently small δ > 0 have a specific arithmetic structure, generalizing the doubling < 4 case.
Eberhard,The abelian arithmetic regularity lemma
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abstract
We give a brief exposition and proof of the arithmetic regularity lemma of Green and Tao in the abelian ($U^2$) case, over $\{1,\dots,N\}$. This may be useful to those who need just the $U^2$ case of the lemma, as the general case is significantly more involved. It may also be useful as an introduction to the general case. No originality is claimed.
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math.NT 3years
2026 3verdicts
UNVERDICTED 3representative citing papers
For dense subsets A of F_p with |A| ≥ αp, max(|A+A|, |A·A|) ≥ (f(α) - o(1))p where f(α) is the optimal constant explicitly determined here.
Arithmetic regularity decomposes arithmetic problems into real, p-adic, and combinatorial factors to obtain correct lower bounds on solution counts in dense sets, illustrated on a linear-plus-higher-degree equation system.
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A structure theorem for sets with doubling $4+\delta$
Integer sets with doubling at most 4 + δ for sufficiently small δ > 0 have a specific arithmetic structure, generalizing the doubling < 4 case.
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For dense subsets A of F_p with |A| ≥ αp, max(|A+A|, |A·A|) ≥ (f(α) - o(1))p where f(α) is the optimal constant explicitly determined here.
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Arithmetic regularity as an alternative to transference
Arithmetic regularity decomposes arithmetic problems into real, p-adic, and combinatorial factors to obtain correct lower bounds on solution counts in dense sets, illustrated on a linear-plus-higher-degree equation system.