Sets of integers that do not contain long arithmetic progressions
The electronic journal of combinatorics, Tome 18 (2011) no. 1
Combining ideas of Rankin, Elkin, Green & Wolf, we give constructive lower bounds for $r_k(N)$, the largest size of a subset of $\{1,2,\dots,N\}$ that does not contain a $k$-element arithmetic progression: For every $\epsilon>0$, if $N$ is sufficiently large, then $$r_3(N) \geq N \left(\frac{6\cdot 2^{3/4} \sqrt{5}}{e \,\pi^{3/2}}-\epsilon\right) \exp\left({-\sqrt{8\log N}+\tfrac14\log\log N}\right),$$ $$r_k(N) \geq N \, C_k\,\exp\left({-n 2^{(n-1)/2} \sqrt[n]{\log N}+\tfrac{1}{2n}\log\log N}\right),$$ where $C_k>0$ is an unspecified constant, $\log=\log_2$, $\exp(x)=2^x$, and $n=\lceil{\log k}\rceil$. These are currently the best lower bounds for all $k$, and are an improvement over previous lower bounds for all $k\neq4$.
@article{10_37236_546,
author = {Kevin O'Bryant},
title = {Sets of integers that do not contain long arithmetic progressions},
journal = {The electronic journal of combinatorics},
year = {2011},
volume = {18},
number = {1},
doi = {10.37236/546},
zbl = {1215.11006},
url = {http://geodesic.mathdoc.fr/articles/10.37236/546/}
}
Kevin O'Bryant. Sets of integers that do not contain long arithmetic progressions. The electronic journal of combinatorics, Tome 18 (2011) no. 1. doi: 10.37236/546
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