2015/10/29 by Ben Green, Green, Ben, Tom Sanders +1
Mathematics · #Advanced Banach Space Theory #Approximation Theory and Sequence Spaces #Combinatorics (math.CO) #FOS: Mathematics #Mathematical Analysis and Transform Methods #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1510.08739
openalex publication_date 2015/10/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
Let \mathbbF be a fixed finite field, and let A ⊂ \mathbbFn. It is a well-known fact that there is a subspace V ≤ \mathbbFn, codim V ≪δ 1, and an x, such that A is δ-uniform when restricted to x + V (that is, all non-trivial Fourier coefficients of A restricted to x + V have magnitude at most δ). We show that if \mathbbF = \mathbbF2 then it is possible to take x = 0; that is, A is δ-uniform on a subspace V ≤ \mathbbFn. We give an example to show that this is not necessarily possible when \mathbbF = \mathbbF3. ADDED July 2016: shortly after this paper appeared on the arxiv, F. Manners showed us a rather short argument he had found in 2013, giving a better bound for our main theorem. We do not, therefore, intend to publish this note. The example over \mathbbF3 may still be of interest to some readers and so we will not withdraw the paper from the arxiv.