Logarithmic bounds for Roth's theorem via almost-periodicity

@article{Bloom2018LogarithmicBF,
  title={Logarithmic bounds for Roth's theorem via almost-periodicity},
  author={Thomas F. Bloom and Olof Sisask},
  journal={discrete Analysis},
  year={2018}
}
Logarithmic bounds for Roth's theorem via almost-periodicity, Discrete Analysis 2019:4, 20pp. A central result of additive combinatorics, Roth's theorem, asserts that for every $\delta>0$ there exists $N$ such that every subset of $\{1,2,\dots,N\}$ of size at least $\delta N$ contains an arithmetic progression of length 3. This is the first non-trivial case of Szemeredi's theorem, proved over 20 years later, which is the corresponding statement for progressions of general length. If we define… 

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