Diameter‐free estimates for the quadratic Vinogradov mean value theorem

Abstract

Let s ⩾ 3 $s \geqslant 3$ be a natural number, let ψ ( x ) $\psi (x)$ be a polynomial with real coefficients and degree d ⩾ 2 $d \geqslant 2$ , and let A $A$ be some large, non-empty, finite subset of real numbers. We use E s , 2 ( A ) $E_{s,2}(A)$ to denote the number of solutions to the system of equations ∑ i = 1 s ( ψ ( x i ) − ψ ( x i + s ) ) = ∑ i = 1 s ( x i − x i + s ) = 0 , \begin{equation*}\hskip2.5pc \sum _{i=1}^{s} (\psi (x_i) - \psi (x_{i+s}) )= \sum _{i=1}^{s} (x_i - x_{i+s} ) = 0, \end{equation*} where x i ∈ A $x_i \in A$ for each 1 ⩽ i ⩽ 2 s $1 \leqslant i \leqslant 2s$ . Our main result shows that E s , 2 ( A ) ≪ d , s | A | 2 s − 3 + η s , \begin{equation*}\hskip6pc E_{s,2}(A) \ll _{d,s} |A|^{2s -3 + \eta _{s}}, \end{equation*} where η 3 = 1 / 2 $\eta _3 = 1/2$ , and η s = ( 1 / 4 − 1 / 7246 ) · 2 − s + 4 $\eta _{s} = (1/4- 1/7246)\cdot 2^{-s + 4}$ when s ⩾ 4 $s \geqslant 4$ . The only other previously known result of this flavour is due to Bourgain and Demeter, who showed that when ψ ( x ) = x 2 $\psi (x) = x^2$ and s = 3 $s=3$ , we have E 3 , 2 ( A ) ≪ ε | A | 3 + 1 / 2 + ε , \begin{equation*}\hskip6pc E_{3,2}(A) \ll _{\epsilon } |A|^{3 + 1/2 + \epsilon }, \end{equation*} for each ε > 0 $\epsilon > 0$ . Thus, our main result improves upon the above estimate, while also generalising it for larger values of s $s$ and more wide-ranging choices of ψ ( x ) $\psi (x)$ . The novelty of our estimates is that they only depend on d $d$ , s $s$ and | A | $|A|$ , and are independent of the diameter of A $A$ . Thus, when A $A$ is a sparse set, our results are stronger than the corresponding bounds that are provided by methods such as decoupling and efficient congruencing. Consequently, our strategy differs from these two lines of approach, and we employ techniques from incidence geometry, arithmetic combinatorics and analytic number theory. Amongst other applications, our estimates lead to stronger discrete restriction estimates for sparse sequences.