Abstract
We give improved bounds for our theorem in [GW09], which shows that a system of linear forms on $\mathbb{F}_p^n$ with squares that are linearly independent has the expected number of solutions in any linearly uniform subset of $\mathbb{F}_p^n$. While in [GW09] the dependence between the uniformity of the set and the resulting error in the average over the linear system was of tower type, we now obtain a doubly exponential relation between the two parameters. Instead of the structure theorem for bounded functions due to Green and Tao [GrT08], we use the Hahn-Banach theorem to decompose the function into a quadratically structured plus a quadratically uniform part. This new decomposition makes more efficient use of the $U^3$ inverse theorem [GrT08].
Original language | English |
---|---|
Pages (from-to) | 121-186 |
Number of pages | 66 |
Journal | Journal d'Analyse Mathématique |
Volume | 115 |
Issue number | 1 |
DOIs | |
Publication status | Published - Jun 2011 |
Bibliographical note
26 pagesKeywords
- math.NT
- math.CO
- 11B30