Arithmetic progressions in sumsets of sparse sets
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Noga Alon
Abstract
A set of positive integers A ⊂ ℤ>0 is log-sparse if there is an absolute constant C so that for any positive integer x the sequence contains at most C elements in the interval [x, 2x). In this note, we study arithmetic progressions in sums of logsparse subsets of ℤ>0. We prove that for any log-sparse subsets S1, . . . , Sn of ℤ>0, the sumset S = S1 + ⋅ ⋅ ⋅ + Sn cannot contain an arithmetic progression of size greater than n(1+o(1))n. We also show that this is nearly tight by proving that there exist log-sparse sets S1, . . . , Sn such that S1 +⋅ ⋅ ⋅+Sn contains an arithmetic progression of size n(1−o(1))n.
Abstract
A set of positive integers A ⊂ ℤ>0 is log-sparse if there is an absolute constant C so that for any positive integer x the sequence contains at most C elements in the interval [x, 2x). In this note, we study arithmetic progressions in sums of logsparse subsets of ℤ>0. We prove that for any log-sparse subsets S1, . . . , Sn of ℤ>0, the sumset S = S1 + ⋅ ⋅ ⋅ + Sn cannot contain an arithmetic progression of size greater than n(1+o(1))n. We also show that this is nearly tight by proving that there exist log-sparse sets S1, . . . , Sn such that S1 +⋅ ⋅ ⋅+Sn contains an arithmetic progression of size n(1−o(1))n.
Kapitel in diesem Buch
- Frontmatter I
- Foreword V
- Contents VII
- Multivariate difference Gončarov polynomials 1
- On an inequality in a 1970 paper of R. L. Graham 21
- Arithmetic progressions in sumsets of sparse sets 27
- Multidimensional Padé approximation of binomial functions: equalities 35
- Graphical enumeration and stained glass windows, 1: rectangular grids 65
- Two extensions of Hilbert’s cube lemma 99
- The Gallai–Ramsey number for a tree versus complete graphs 109
- On Levine’s notorious hat puzzle 115
- Recurrence ranks and moment sequences 167
- On weak twins and up-and-down subpermutations 187
- Distance graphs and arithmetic progressions 203
- Consecutive primes which are widely digitally delicate 209
- Spanning trees and domination in hypercubes 249
- Rook domination on hexagonal hexagon boards 259
- Strongly image partition regular matrices 267
- Introducing shift-constrained Rado numbers 285
- Mertens’ prime product formula, dissected 297
- Curious convergent series of integers with missing digits 311
- A note on Carmichael numbers in residue classes 321
- Tilted corners in integer grids 329
- Remembrances 339
- A selected bibliography of Ron Graham 355
Kapitel in diesem Buch
- Frontmatter I
- Foreword V
- Contents VII
- Multivariate difference Gončarov polynomials 1
- On an inequality in a 1970 paper of R. L. Graham 21
- Arithmetic progressions in sumsets of sparse sets 27
- Multidimensional Padé approximation of binomial functions: equalities 35
- Graphical enumeration and stained glass windows, 1: rectangular grids 65
- Two extensions of Hilbert’s cube lemma 99
- The Gallai–Ramsey number for a tree versus complete graphs 109
- On Levine’s notorious hat puzzle 115
- Recurrence ranks and moment sequences 167
- On weak twins and up-and-down subpermutations 187
- Distance graphs and arithmetic progressions 203
- Consecutive primes which are widely digitally delicate 209
- Spanning trees and domination in hypercubes 249
- Rook domination on hexagonal hexagon boards 259
- Strongly image partition regular matrices 267
- Introducing shift-constrained Rado numbers 285
- Mertens’ prime product formula, dissected 297
- Curious convergent series of integers with missing digits 311
- A note on Carmichael numbers in residue classes 321
- Tilted corners in integer grids 329
- Remembrances 339
- A selected bibliography of Ron Graham 355
