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

Abstract

Abstract. The problem of looking for subsets of the natural numbers which contain no 3-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-progressionfree set. We also show that geometric progressions in Z/nZ 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. BackgroundLet A be a subset of the positive integers. A three-term arithmetic progression in A is a progression (a, a + b, a + 2b) ∈ A 3 with b > 0, or equivalently, a solution to the equation a + c = 2b where a, b and c are distinct elements of A. We say that A is free of arithmetic progressions if it contains no such progressions. For any subset A of the positive integers we denote by d(A) its asymptotic density (if it exists) and its upper density byd(A).In 1952 Roth proved the following famous theorem [14]. Theorem 1.1 (Roth).If A is a subset of the positive integers withd(A) > 0 then A contains a 3-term arithmetic progression.In particular, Roth showed that for any fixed α > 0 and sufficiently large N, any subset of the integers {1, · · · , N} of size at least αN contains a 3-term arithmetic progression. We can also view Roth's result as a statement about arithmetic progressions (with 3 distinct elements) in the group of integers mod N. Namely, if we denote by D(Z/NZ) the size of the largest subset of Z/NZ free of arithmetic progressions, Roth's argument can be used to show Roth's theorem has since been generalized by Szemerédi to progressions of arbitrary length. Arithmetic-progression-free sets have also been studied in the context of arbitrary abelian groups. Meshulam [9] generalized Roth's theorem to finite abelian groups of odd order, and recently Lev [8] has extended Meshulam's ideas to arbitrary finite abelian groups, a result which will be needed later. Let D(G) be the size of the largest subset

Geometric-progression-free sets over quadratic number fields

“…In the case where the smallest non-unit element has norm 3, we have that the proportion of excluded elements is bounded above by (4.4) and in the case where the smallest non-unit element has norm 4, we have that the proportion of excluded elements is bounded above by We can conceivably improve these bounds by taking into account more possible ratios. We do so, generalizing an argument made over the integers in [4]. For a quadratic number field K = Q √ d , and an integer n ∈ Z, we define an element (or ideal) of the ring of integers O K to be n-smooth if its factorization into irreducible elements consists entirely of elements with norm at most n.…”

“…The results of this computation are summarized in Table 4. [4], also discussed more recently by Nathanson and O'Bryant [7]. For illustration, we first consider the Gaussian integers and construct a subset S ⊂ Z[i] that has a large upper density and that avoids geometric progressions.…”

“…Rankin constructed a larger set, with asymptotic density approximately 0.71974. Modifying the construction of this set slightly, (see [4]) one can get a small improvement, producing a set with density about 0.72195. It is not known what the greatest possible asymptotic density of such a set is, however several authors have considered sets with greater upper asymptotic density.…”

Subsets of Fq[x] free of 3-term geometric progressions

“…It seems possible that a similar construction might yield a set with higher asymptotic density in F q [x]. In previous work, [Mc,Section 4], [BHMMPTW,Section 4.2] in different rings, improved upper bounds for the upper density were obtained by considering progressions among the smooth integers. Thus far this technique has not proven to be as useful in this ring, however it may just be that more work and computation are required.…”

Geometric progression-free sequences with small gaps

“…Let A be a k-GP-free sequence. The question of finding the maximal possible asymptotic density d(A), or else the upper density d U (A), of A has been a subject of recent study [2,3,12,15,15]. For an exposition of progress on this problem and the tightest known bounds on d U (A), along with constructions of A with nearly optimal upper density, see the paper of McNew [12].…”