7. Near Arithmetic Progressions
Given $t,d\in\mathbb{N}$ and $w\in\mathbb{N}_0$, a $(t,d,w)$progression is a set of the form \[ B(b,t,d,w):=\bigcup_{i=0}^{t1}[b+id,b+id+w] \] for some $b\in\mathbb{N}_0$. A set $A\subseteq\mathbb{N}$ nearly contains a $(t,d,w)$progression if there is a $(t,d,w)$progression $B(b,t,d,w)$ such that $A\cap [b+id,b+id+w]\neq\emptyset$ for all $i=1,\ldots,t1$.Given $t\in\mathbb{N}$, let $g_t(n)$ be the largest element of $\{0,\frac{1}{n},\frac{2}{n},\ldots,1\}$ such that there is $A\subseteq[1,n]$ satisfying:
 $\frac{A\cap[1,n]}{n}\geq g_t(n)$, and
 $A$ does not nearly contain a $(t,d,w)$progression, where $d$ is a power of $2$ and $\frac{w}{d}<\frac{d}{n}$.

A positive answer to the following conjecture would imply the ErdősTurán conjecture: if $A\subseteq\mathbb{N}$ is such that $\sum_{a\in A}\frac{1}{a}$ diverges, then $A$ contains arbitrarily long finite arithmetic progressions (see chapter 15 of the workshop book).
Conjecture 7.1.
[Steven Leth] For all $t\in\mathbb{N}$ and sufficiently large $n$, \[ g_t(n)<\frac{1}{(\log n)^{\log \log n}}. \] 
It is known that for every $t\in\mathbb{N}$ and $\epsilon>0$, $g_t(n)>\frac{1}{n^\epsilon}$ for sufficiently large $n$. The following is a weaker form of the previous conjecture, which appears to still be interesting and is unresolved.
Conjecture 7.2.
[Steven Leth] For all $t,k\in\mathbb{N}$ and sufficiently large $n$, \[ g_t(n)<\frac{1}{(\log n)^k}. \]
Cite this as: AimPL: Nonstandard methods in combinatorial number theory, available at http://aimpl.org/nscombinatorial.