Linear forms and quadratic uniformity for functions on ${\mathbb{F}}_p^n$

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].

26 pages

References in corpus (1)

• An equivalence between inverse sumset theorems and inverse conjectures for the U^3 norm

Cited by in corpus (5)

• On the polynomial Szemerédi theorem in finite fields
• Linear forms and quadratic uniformity for functions on ${\mathbb{Z}}_N$
• Induced arithmetic removal: complexity 1 patterns over finite fields
• True complexity of polynomial progressions in finite fields
• Local aspects of the Sidorenko property for linear equations