arXiv:2007.03528math.NT
Breaking the logarithmic barrier in Roth's theorem on arithmetic progressions
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Abstract
We show that if $A\subset \{1,\ldots,N\}$ contains no non-trivial three-term arithmetic progressions then $\lvert A\rvert \ll N/(\log N)^{1+c}$ for some absolute constant $c>0$. In particular, this proves the first non-trivial case of a conjecture of Erdős on arithmetic progressions.
The conjecture by Erdős is the following: if A ⊂ ℕ is such that Σ_{n ∈ A} 1/n diverges, then A contains arbitrarily long arithmetic progressions. Bloom and Sisask now proved that A contains infinitely many length-3 arithmetic progressions, following from their main result which…
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