Sets whose sumset avoids a thin sequence

Abstract

Let {a 1 , a 2 , a 3 , . . .} be an unbounded sequence of positive integers with a n+1 /a n approaching α as n → ∞, and let β > max(α, 2). We show that for all sufficiently large xa set of nonnegative integers containing 0 and satisfying |A| 1 − 1 β x, then we can represent some element of the sequence {a n } as a pairwise sum of elements of A. We also prove an analogous result which holds for all x 0. In [1], Erdős and Freud conjectured if A ⊂ {1, 2, . . . , 3n} is a set of at least n + 1 elements then there is some power of two that could be written as the sum of distinct elements of A. This was proved by Erdős and Freiman [2] for sufficiently large n, and later improved by Nathanson and Sárközy [4]. We could also ask what happens if we restrict the number of summands. Lev [3] showed that if A ⊂ {0, 1, 2, . . . ,n} contains 0 and has at least n/2 + 1 elements, then a power of 2 can be written as the sum of two elements of A. And more recently, Pan [5] proved the essentially sharp result (for m 3): if 0 ∈ A, then |A| (1 − 1/m)n + 1 implies that a power of m can be written as the sum of two elements in A.In this paper, we consider sequences that grow essentially like the powers of a real number greater than or equal to 2. Given a set A, denote by 2A the set of pairwise sums {a 1 + a 2 : a 1 , a 2 ∈ A}. (Note that this set is usually denoted by A + A.) We then have the following result:

On the cardinality of general h-fold sumsets

“…Then X g → X g+1 is a (r, h)-path. Since X g satisfies (1), it follows that X g+1 also satisfies (1). This is a contradiction with the maximality of g.…”

“…. , x g,k−1 ), then X g → X g+1 is a (r, h)-path and X g+1 also satisfies (1). This is a contradiction with the maximality of g. Hence x g,s−1 = 0.…”

Sumsets Containing a Term of a Sequence

“…In 2010, Kapoor [3] extended Pan's result for 2A to general sequences. He proved the following two results.…”

Sumsets and Difference Sets Containing a Common Term of a Sequence

“…For a set A of integers, let A + A = {a 1 + a 2 : a 1 , a 2 ∈ A} and A − A = {a 1 − a 2 : a 1 , a 2 ∈ A}. Recently, Kapoor [5] extended the above results to general sequences. He proved that for an unbounded sequence {a k } of positive integers with a k+1 /a k → α as k → ∞, and β > max(α, 2), if A ⊂ [0, x] is a set of integers with 0 ∈ A and…”