Diameter?free estimates for the quadratic Vinogradov mean value theorem
نویسندگان
چکیده
Let s ? 3 $s \geqslant 3$ be a natural number, let ? ( x ) $\psi (x)$ polynomial with real coefficients and degree d 2 $d 2$ , A $A$ some large, non-empty, finite subset of numbers. We use E $E_{s,2}(A)$ to denote the number solutions system equations ? i = 1 ? + 0 \begin{equation*}\hskip2.5pc \sum _{i=1}^{s} (\psi (x_i) - \psi (x_{i+s}) )= (x_i x_{i+s} 0, \end{equation*} where ? $x_i \in A$ for each ? $1 \leqslant 2s$ . Our main result shows that ? | ? \begin{equation*}\hskip6pc E_{s,2}(A) \ll _{d,s} |A|^{2s -3 \eta _{s}}, / $\eta _3 1/2$ 4 7246 · _{s} (1/4- 1/7246)\cdot 2^{-s 4}$ when 4$ The only other previously known this flavour is due Bourgain Demeter, who showed (x) x^2$ $s=3$ we have ? E_{3,2}(A) _{\epsilon } |A|^{3 1/2 \epsilon }, > $\epsilon 0$ Thus, our improves upon above estimate, while also generalising it larger values $s$ more wide-ranging choices novelty estimates they depend on $d$ $|A|$ are independent diameter sparse set, results stronger than corresponding bounds provided by methods such as decoupling efficient congruencing. Consequently, strategy differs from these two lines approach, employ techniques incidence geometry, arithmetic combinatorics analytic theory. Amongst applications, lead discrete restriction sequences.
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ژورنال
عنوان ژورنال: Proceedings of The London Mathematical Society
سال: 2022
ISSN: ['1460-244X', '0024-6115', '1234-5678']
DOI: https://doi.org/10.1112/plms.12489