On sets of integers which contain no three terms in geometric progression

Abstract:

The problem of looking for subsets of the natural numbers which contain no three-term arithmetic progressions has a rich history. Roth’s theorem famously shows that any such subset cannot have positive upper density. In contrast, Rankin in 1960 suggested looking at subsets without three-term geometric progressions, and constructed such a subset with density about 0.719. More recently, several authors have found upper bounds for the upper density of such sets. We significantly improve upon these bounds, and demonstrate a method of constructing sets with a greater upper density than Rankin’s set. This construction is optimal in the sense that our method gives a way of effectively computing the greatest possible upper density of a geometric-progression-free set. We also show that geometric progressions in $\mathbb {Z}/n\mathbb {Z}$ behave more like Roth’s theorem in that one cannot take any fixed positive proportion of the integers modulo a sufficiently large value of $n$ while avoiding geometric progressions.