Geometric progressions in syndetic sets
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Archiv der Mathematik
Geometric progressions in syndetic sets Melvyn B. Nathanson
Abstract. An open problem about finite geometric progressions in syndetic sets leads to a family of diophantine equations related to the commutativity of translation and multiplication by squares. Mathematics Subject Classification. 05D10, 11B75, 11D09, 11R11. Keywords. Geometric progressions, Syndetic sets, Diophantine equations, Pell’s equation.
1. The problem of geometric progressions in syndetic sets. The set A = {ai : i = 1, 2, 3, . . .} of positive integers is strictly increasing if a1 < a2 < a3 < · · · . The set A has gaps bounded by if ai+1 −ai ≤ for all i = 1, 2, 3, . . .. The set A has bounded gaps if it has gaps bounded by for some . A set with bounded gaps is also called syndetic. Beiglb¨ ock et al. [1] asked if every syndetic set of integers contains arbitrarily long finite geometric progressions. For recent progress on this still unsolved problem, see Glasscock et al. [2] and Patil [7]. For sets not containing finite geometric progressions, see McNew [3] and Nathanson and O’Bryant [4–6]. It is not even known if a syndetic set contains a three-term geometric progression, that is, a subset of the form {a, ax, ax2 } for integers a and x with x ≥ 2. Indeed, it is not known if a syndetic set must contain a subset of the form {a, ax2 } for some integer x ≥ 2. Patil [7] recently proved the following special case: Every strictly increasing set of positive integers with gaps bounded by 2 contains infinitely many subsets {a, ax2a } with xa ≥ 2. His proof uses the following beautiful polynomial identity: a(4a + 3)2 + 1 = (a + 1)(4a + 1)2 .
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This research was supported in part by the International Centre for Theoretical Sciences (ICTS) during a visit for the program—Workshop on Additive Combinatorics (Code: ICTS/Prog-wac2020/02).
M. B. Nathanson
Arch. Math.
The core of the argument is the following result. Theorem 1 (Patil). Let A be a set of positive integers such that {a, a + 1} ⊆ A for infinitely many a ∈ A, and {a, a + 1} ∩ A = ∅ for all positive integers a. The set A contains infinitely many subsets of the form {a, ax2a } with integers xa ≥ 2. The object of this note is to construct an infinite family of arithmetic identities that will generalize Theorem 1. 2. Translations and dilations by squares. Let a, k, , m, and n be positive integers with m ≥ 2 and n ≥ 2. Does there exist an integer b, or do there exist infinitely many integers b, such that (i) a multiplied by an mth power and translated by k is equal to b, and (ii) a translated by and multiplied by an nth power is also equal to b? Thus, the problem asks about the existence of a pair of positive integers (x, y), or of infinitely many pairs of positive integers (x, y), such that axm + k = (a + )y n . A necessary condition is k ≡ 0 (mod gcd(a, )). We solve this problem in the case of equal translations and of dilations by squares, that is, k = and m = n = 2. Theorem 2. Let a and k be positive integers, and let d = a(a + k). Let (u, v) be a positive inte
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