Arithmetic structures in smooth subsets of F_p
| dc.creator | Croot, Ernie | |
| dc.date | 2007-08-27 | |
| dc.date.accessioned | 2026-07-07T08:26:03Z | |
| dc.date.available | 2026-07-07T08:26:03Z | |
| dc.description | Fix integers a_1,...,a_d satisfying a_1 + ... + a_d = 0. Suppose that f : Z_N -> [0,1], where N is prime. We show that if f is ``smooth enough'' then we can bound from below the sum of f(x_1)...f(x_d) over all solutions (x_1,...,x_d) in Z_N to a_1 x_1 + ... + a_d x_d == 0 (mod N). Note that d = 3 and a_1 = a_2 = 1 and a_3 = -2 is the case where x_1,x_2,x_3 are in arithmetic progression. By ``smooth enough'' we mean that the sum of squares of the lower order Fourier coefficients of f is ``small'', a property shared by many naturally-occurring functions, among them certain ones supported on sumsets and on certain types of pseudoprimes. The paper can be thought of as a generalization of another result of the author, which dealt with a F_p^n analogue of the problem. It appears that the method in that paper, and to a more limited extent the present paper, uses ideas similar to those of B. Green's ``arithmetic regularity lemma'', as we explain in the paper. | |
| dc.description | This is a very preliminary draft. Future drafts will have cleaner proofs and tighter notation | |
| dc.identifier | https://arxiv.org/abs/0708.3689 | |
| dc.identifier | http://arxiv.org/abs/0708.3689 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/136819 | |
| dc.subject | Number Theory | |
| dc.subject | Combinatorics | |
| dc.subject | 11P99 | |
| dc.title | Arithmetic structures in smooth subsets of F_p | |
| dc.type | text |