On the Littlewood problem modulo a prime
| dc.creator | Green, Ben | |
| dc.creator | Konyagin, Sergei | |
| dc.date | 2006-01-23 | |
| dc.date | 2006-06-19 | |
| dc.date.accessioned | 2026-07-07T06:59:12Z | |
| dc.date.available | 2026-07-07T06:59:12Z | |
| dc.description | Let p be a prime, and let f : Z/pZ -> R be a function with average value 0 and ||f||_A <= 1, where ||f||_A denotes the algebra norm (L^1 norm of the Fourier transform). Then f(x) is small for some x, specifically min_x |f(x)| is no more than O(log p)^{-1/3 + eps}. One should think of f as being ``approximately continuous''; our result is then an ``approximate intermediate value theorem''. As an immediate consequence we show that if B in Z/pZ is a set of cardinality (p-1)/2 then the algebra norm ||1_B||_A is >> (log p)^{1/3 - eps}. This gives a result on a ``mod p'' analogue of Littlewood's well-known problem concerning the smallest possible L^1-norm of the Fourier transform of a set of n integers. Another application is to answer a question of Gowers. If B in Z/pZ is a set of size (p-1)/2 then there is some x in Z/pZ such that the intersection of B with B + x has cardinality within o(p) of p/4. | |
| dc.description | 23 pages, to appear in Canadian J. Math | |
| dc.identifier | https://arxiv.org/abs/math/0601565 | |
| dc.identifier | http://arxiv.org/abs/math/0601565 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/107663 | |
| dc.subject | Classical Analysis and ODEs | |
| dc.subject | Number Theory | |
| dc.title | On the Littlewood problem modulo a prime | |
| dc.type | text |