On sets of integers not containing long arithmetic progressions

Izabella Laba; Michael T. Lacey

On sets of integers not containing long arithmetic progressions

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arxiv math/0108155 v1 pith:LX3WOD5H submitted 2001-08-22 math.CO math.NT

Izabella Laba , Michael T. Lacey This is my paper · ORCID

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keywords progressionsarithmeticcontainlengthsetsaritmeticbehrendcardinality

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We construct subsets of {1,...,N} of cardinality at least N exp(-C(log N)^{1/(k+1)}) which do not contain arithmetic progressions of length 2^k+1. This extends a result of Behrend (1946) concerning sets which do not contain aritmetic progressions of length 3.

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Counting subsets of integers free of arithmetic configurations
math.CO 2026-07 conditional novelty 8.0 of 10

For k≥5, infinitely many n have exactly 2^{r_k(n)(1+o(1))} k-AP-free subsets of [n]; for all n and k≥3 the count is 2^{O(r_k(n))}.

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