论文标题
sárközy对二次残基的猜想，ii
A conjecture of Sárközy on quadratic residues, II
论文作者
论文摘要
用$ \ Mathcal {r} _p $表示$ \ Mathbf {f} _p $的所有二次残基的集合，每个prime $ p $。对于所有足够大的$ p $，a.sárközy的猜想都没有子集$ \ mathcal {a}，\ m nathcal {b} \ subseteq \ subseteq \ mathbf {f} _p {f} _p { $ \ MATHCAL {A}+\ MATHCAL {B} = \ MATHCAL {R} _p $。在本文中，我们表明，如果此类子集$ \ MATHCAL {a}，\ MATHCAL {B} $确实存在，那么至少存在$（\ log 2）^{ - 1} \ sqrt p-1.6 $元素，$ \ \ \ \ \ \ \ m natercal {a}+\ \ \ \ \ \ \ \ \ \ nsiment a an \ frac {1} {4} \ sqrt {p} <| \ mathcal {a} |，| \ Mathcal {b} | <2 \ 2 \ sqrt {p} -1。 \ end {Align*}这可以完善由I.E.获得的先前界限。 Shparlinski，I.D。 Shkredov和Y.-G。陈和X.-H。 YAN。此外，我们还为$ | \ Mathcal {a} |，| \ Mathcal {b} | $建立了界限，以及添加的能量$ e（\ Mathcal {a}，\ Mathcal {b}）$，如果$ \ Mathcal {a}+\ \ \ \ \ \ m natercal {a}+\ m nresscal {a}
Denote by $\mathcal{R}_p$ the set of all quadratic residues in $\mathbf{F}_p$ for each prime $p$. A conjecture of A. Sárközy asserts, for all sufficiently large $p$, that no subsets $\mathcal{A},\mathcal{B}\subseteq\mathbf{F}_p$ with $|\mathcal{A}|,|\mathcal{B}|\geqslant2$ satisfy $\mathcal{A}+\mathcal{B}=\mathcal{R}_p$. In this paper, we show that if such subsets $\mathcal{A},\mathcal{B}$ do exist, then there are at least $(\log 2)^{-1}\sqrt p-1.6$ elements in $\mathcal{A}+\mathcal{B}$ that have unique representations and one should have \begin{align*} \frac{1}{4}\sqrt{p}< |\mathcal{A}|,|\mathcal{B}|< 2\sqrt{p}-1. \end{align*} This refines previous bounds obtained by I.E. Shparlinski, I.D. Shkredov, and Y.-G. Chen and X.-H. Yan. Moreover, we also establish bounds for $|\mathcal{A}|,|\mathcal{B}|$ and the additive energy $E(\mathcal{A},\mathcal{B})$ if few elements in $\mathcal{A}+\mathcal{B}$ have unique representations.
