Sets of Integers that do not Contain Long Arithmetic Progressions
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 $k\neq4$.
1999 ◽
Vol 60
(1)
◽
pp. 21-35
2014 ◽
Vol 57
(3)
◽
pp. 551-561
◽
2008 ◽
Vol 51
(1)
◽
pp. 47-56
◽
Keyword(s):
Keyword(s):
1968 ◽
Vol 11
(3)
◽
pp. 409-414
◽
2009 ◽
Vol 05
(04)
◽
pp. 625-634
Keyword(s):
2008 ◽
Vol 78
(3)
◽
pp. 431-436
◽