论文标题
关于最大产品集的随机组集
On maximal product sets of random sets
论文作者
论文摘要
对于[0,1）$中的每个正整数n和每个$α\，让$ b（n，α）$表示概率模型，其中随机设置$ a \ subset \ a \ subset \ {1，\ dots，n \} $是通过独立选择$ \ \ dots的每个元素而构建的。我们证明，作为$ n \ longrightArrow +\ infty $，对于$ b（n，α）$中的每一个$ a $ a $ a $ | aa | aa | \ sim | \ sim | a |^2/2 $带有概率$ 1-o（1）$，if的且仅当$ $ \ frac \ frac \ frac {\ frac {\ log log（α^2（α^2（α^2）） 4-1}）}}} {\ sqrt {\ log \ log n}} \ longrightArrow- \ infty。$$这改善了Cilleruelo，Ramana和Ramaré的定理，他们证明了上述$ | AA | aa | aa | a | a | a | a | a | a | a | a | a | a |^2/2 $ n $ n $ n = o（$ n = o（1/2）最大产品集的完整表征。
For every positive integer N and every $α\in [0,1)$, let $B(N, α)$ denote the probabilistic model in which a random set $A\subset \{1,\dots,N\}$ is constructed by choosing independently every element of $\{1,\dots,N\}$ with probability $α$. We prove that, as $N\longrightarrow +\infty$, for every $A$ in $B(N, α)$ we have $|AA|\ \sim |A|^2/2$ with probability $1-o(1)$, if and only if $$\frac{\log(α^2(\log N)^{\log 4-1})}{\sqrt{\log\log N}}\longrightarrow-\infty.$$ This improves a theorem of Cilleruelo, Ramana and Ramaré, who proved the above asymptotic between $|AA|$ and $|A|^2/2$ when $α=o(1/\sqrt{\log N})$, and supplies a complete characterization of maximal product sets of random sets.
