MATH Seminar
| Title: Problems on Sidon sets of integers |
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| Seminar: Dissertation Defense |
| Speaker: Sangjune Lee of Emory University |
| Contact: Sangjune Lee, slee242@emory.edu |
| Date: 2012-04-03 at 2:30PM |
| Venue: W304 |
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| Abstract: A set~$A$ of non-negative integers is a \textit{Sidon set} if all the sums~$a_1+a_2$, with~$a_1\leq a_2$ and~$a_1$,~$a_2\in A$, are distinct. In this dissertation, we deal with three results on Sidon sets: two results are about finite Sidon sets in $[n]=\{0,1,\cdots, n-1\}$ and the last one is about infinite Sidon sets in $\mathbb{N}$ (the set of natural numbers). \\ \\ First, we consider the problem of Cameron--Erd\H{o}s estimating the number of Sidon sets in $[n]$. We obtain an upper bound $2^{c\sqrt{n}}$ on the number of Sidon sets which is sharp with the previous lower bound up to a constant factor in the exponent. \\ \\ Next, we study the maximum size of Sidon sets contained in sparse random sets $R\subset [n]$. Let~$R=[n]_m$ be a uniformly chosen, random $m$-element subset of~$[n]$. Let $F([n]_m)=\max\{|S|\colon S\subset[n]_m\hbox{ is Sidon}\}$. Fix a constant~$0\leq a\leq1$ and suppose~$m=(1+o(1))n^a$. We show that there is a constant $b=b(a)$ for which~$F([n]_m)=n^{b+o(1)}$ almost surely and we determine $b=b(a)$. Surprisingly, between two points $a=1/3$ and $a=2/3$, the function~$b=b(a)$ is constant. \\ \\ Next, we deal with infinite Sidon sets in sparse random subsets of $\mathbb{N}$. Fix $0<\delta\leq 1$, and let $R=R_{\delta}$ be the set obtained by choosing each element $i\subset\mathbb{N}$ independently with probability $i^{-1+\delta}$. We show that for every $0<\delta\leq 2/3$ there exists a constant $c=c(\delta)$ such that a random set $R$ satisfies the following with probability 1: \begin{itemize} \item Every Sidon set $S\subset R$ satisfies that $|S\cap [n]|\leq n^{c+o(1)}$ for every sufficiently large $n$. \item There exists a large Sidon set $S\subset R$ such that $|S\cap [n]| \geq n^{c+o(1)}$ for every sufficiently large $n$. \end{itemize} |
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