Three-term polynomial progressions in subsets of finite fields

Abstract

Bourgain and Chang recently showed that any subset of Fp of density ≫ p −1/15 contains a nontrivial progression x, x + y, x + y 2 . We answer a question of theirs by proving that if P1, P2 ∈ Z[y] are linearly independent and satisfy P1(0) = P2(0) = 0, then any subset of Fp of density ≫P 1 ,P 2 p −1/24 contains a nontrivial polynomial progression x, x + P1(y), x + P2(y).

On the polynomial Szemerédi theorem in finite fields

“…The dependence is extremely poor, so we do not keep track of it. We also remark that while the earlier papers [13] and [4] both rely on a decent amount of algebraic geometry machinery, the proof of Theorem 1.1 only requires the Weil bound for curves.…”

“…Bourgain and Chang's proof was quite specific to the progression x, x + y, x + y 2 , and relied on the explicit evaluation of quadratic Gauss sums. Using a different argument, the author showed in [13] that a result like Bourgain and Chang's holds when y and y 2 are replaced by any two linearly independent polynomials P 1 and P 2 with P 1 (0) = P 2 (0) = 0. The main result of [13] is that (3) #{(x, y) ∈ F 2 q : x, x + P 1 (y), x + P 2 (y) ∈ A} = |A| 3 q + O P 1 ,P 2 (|A| 3/2 q 7/16 ) for any A ⊂ F q whenever the characteristic of F q is sufficiently large, so that r P 1 ,P 2 (F q ) ≪ P 1 ,P 2 q 1−1/24 .…”

“…Using a different argument, the author showed in [13] that a result like Bourgain and Chang's holds when y and y 2 are replaced by any two linearly independent polynomials P 1 and P 2 with P 1 (0) = P 2 (0) = 0. The main result of [13] is that (3) #{(x, y) ∈ F 2 q : x, x + P 1 (y), x + P 2 (y) ∈ A} = |A| 3 q + O P 1 ,P 2 (|A| 3/2 q 7/16 ) for any A ⊂ F q whenever the characteristic of F q is sufficiently large, so that r P 1 ,P 2 (F q ) ≪ P 1 ,P 2 q 1−1/24 . Note that the exponent of q in the error term of (3) is larger than in (2), so that the argument in [13] does not quantitatively recover the result in [3].…”

“…The main result of [13] is that (3) #{(x, y) ∈ F 2 q : x, x + P 1 (y), x + P 2 (y) ∈ A} = |A| 3 q + O P 1 ,P 2 (|A| 3/2 q 7/16 ) for any A ⊂ F q whenever the characteristic of F q is sufficiently large, so that r P 1 ,P 2 (F q ) ≪ P 1 ,P 2 q 1−1/24 . Note that the exponent of q in the error term of (3) is larger than in (2), so that the argument in [13] does not quantitatively recover the result in [3]. However, this exponent of q does not depend at all on P 1 or P 2 , so the bound r P 1 ,P 2 (F q ) ≪ P 1 ,P 2 q 1−1/24 is stronger than what can possibly hold in the integer setting when at least one of P 1 or P 2 has degree at least 25.…”

“…This also improves on the error term in Bourgain and Chang's result. The arguments in [3], [13], and [4] break down when one tries to use them to study progressions of length longer than three. Currently no results are known for progressions of length at least four when P 1 , .…”