On the number of popular differences
- 82 Downloads
We prove that there exists an absolute constant c > 0 such that for any finite set A ⊆ ℤ with |A| ≥ 2 and any positive integer m < c|A|/ ln |A| there are at most m integers b > 0 satisfying |(A + b) \ A| ≤ m; equivalently, there are at most m positive integers possessing |A| −m (or more) representations as a difference of two elements of A.
This is best possible in the sense that for each positive integer m there exists a finite set A ⊆ ℤ with |A| > m log2(m/2) such that |(A+b)\A| ≤ m holds for b = 1, ..., m + 1.
KeywordsPositive Integer Abelian Group Group Element Pairwise Disjoint Arithmetic Progression
Unable to display preview. Download preview PDF.