论文标题
在整个整数中找到大型添加剂和乘法sidon集
Finding large additive and multiplicative Sidon sets in sets of integers
论文作者
论文摘要
给定$ h，g \ in \ mathbb {n} $，我们编写了一个$ x \ subset \ subset \ mathbb {z} $，为$ b_ {h}^{ +} $设置，如果为任何$ n \ in \ mathbb {z} $ in \ mathbb {z} $，添加$ n = $ n y = x y = x y = x y = x y = x y = y = x y = x y = x y = x y = x y = x y = x y = x y = \ dots，x_h \ in x $中最多是$ g $，如果仅在summands的订购方面有所不同，我们认为两个这样的解决方案是相同的。我们定义一个乘法$ b_ {h}^{\ times} [g] $类似地设置。在本文中，我们证明，除其他结果外，存在$ g \ in \ mathbb {n} $和$δ> 0 $，因此对于任何$ h \ in \ athbb {n} $中的任何$ h \美元，| c | \} \ gg_ {h} | a |^{（1+Δ）/h}。 \]实际上，当$ h = 2 $时，我们可能会设置$ g = 31 $，而当$ h $足够大时，我们可能会设置$ g = 1 $和$δ\ gg（\ log \ log \ log \ log h）^{1/2-o（1）} $。前者朝着最近的猜想迈进了库尔曼 - 帕奥托（Pohoata），并定量加强了什克雷夫（Shkredov）的先前工作。
Given $h,g \in \mathbb{N}$, we write a set $X \subset \mathbb{Z}$ to be a $B_{h}^{+}[g]$ set if for any $n \in \mathbb{Z}$, the number of solutions to the additive equation $n = x_1 + \dots + x_h$ with $x_1, \dots, x_h \in X$ is at most $g$, where we consider two such solutions to be the same if they differ only in the ordering of the summands. We define a multiplicative $B_{h}^{\times}[g]$ set analogously. In this paper, we prove, amongst other results, that there exist absolute constants $g \in \mathbb{N}$ and $δ>0$ such that for any $h \in \mathbb{N}$ and for any finite set $A$ of integers, the largest $B_{h}^{+}[g]$ set $B$ inside $A$ and the largest $B_{h}^{\times}[g]$ set $C$ inside $A$ satisfy \[ \max \{ |B| , |C| \} \gg_{h} |A|^{(1+ δ)/h }. \] In fact, when $h=2$, we may set $g = 31$, and when $h$ is sufficiently large, we may set $g = 1$ and $δ\gg (\log \log h)^{1/2 - o(1)}$. The former makes progress towards a recent conjecture of Klurman--Pohoata and quantitatively strengthens previous work of Shkredov.
