A Structure Theorem for Positive Density Sets Having the Minimal Number of 3-term Arithmetic Progressions
| dc.creator | Croot, Ernie | |
| dc.date | 2003-05-22 | |
| dc.date.accessioned | 2026-07-07T04:58:11Z | |
| dc.date.available | 2026-07-07T04:58:11Z | |
| dc.description | Assuming the well-known conjecture that [x,x+x^t] contains a prime for t > 0 and x sufficiently large, we prove: For 0 < r < 1, there exists 0 < s < r < 1, 0 < d < 1, and infinitely many primes q such that if S is a subset of Z/qZ having density at least s, and having the least number of 3-term arithemtic progressions among all sets of density at least s, then S is nearly translation invariant in a very strong sense. Namely, there exists 0 <= b <= q-1 such that |S intersect (S + bj)| = (1-g(s))|S|, for every 0 < j < q^d, where g(s) -> 0 as s -> 0. A curious feature of the proof is that Behrend's construction on large subsets of {1,2,...,x} containing no 3-term a.p., is a key ingredient. | |
| dc.identifier | https://arxiv.org/abs/math/0305318 | |
| dc.identifier | http://arxiv.org/abs/math/0305318 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/67536 | |
| dc.subject | Number Theory | |
| dc.subject | Combinatorics | |
| dc.subject | 11P70 | |
| dc.title | A Structure Theorem for Positive Density Sets Having the Minimal Number of 3-term Arithmetic Progressions | |
| dc.type | text |