50 problems

Let $f(N)$ be the maximum size of a Sidon subset of $[N]=\{1,\ldots,N\}$, and let $A(N)$ be the number of Sidon subsets of $[N]$. Does $A(N)/2^{f(N)}\to\infty$? Is…

The assertion is false: it is not true that every Sidon set $A\subseteq\mathbb{N}$ has an infinite arithmetic progression $Y\subseteq\mathbb{N}$ with…

Let $r\geq 2$ and let $A\subseteq \{1,\ldots,N\}$ be a set of maximal size such that there are at most $r$ solutions to $n=a+b$ with $a\leq b$ for any $n$. (That is, $A$ is a…

For any $M\geq 1$, if $A\subset \mathbb{N}$ is a sufficiently large finite Sidon set then there are at least $M$ many $a\in A+A$ such that $a+1,a-1 ot\in A+A$.

More generally: Let $a_1 < a_2 < \cdots < a_{k_1} \leq n$, $b_1 < b_2 < \cdots < b_{k_2} \leq n$ be two Sidon sequences for which $a_j - a_i \neq b_v - b_u$ for all $i < j$ and…

Let $1 \leq a_1 < a_2 < \cdots < a_h \leq n$ be a maximum Sidon sequence. Can one find a Sidon sequence $b_1 < b_2 < \cdots < b_r \leq n$ for every $r$ and $n > n_0(r)$ so that the…

Let $m\geq 2$, let $N_k\to\infty$ be a sequence of natural numbers, and let $A_k\subseteq\{0,\ldots,N_k\}$ be Sidon sets, meaning that equal sums of two elements determine the same…

Is there a constant $c>0$ such that, for every $N$, the cubes $1^3,2^3,…,N^3$ contain a Sidon subset of size at least $cN$? Equivalently in the infinite direction, is there a posit…

Let $A=\{a_1<a_2<\cdots\}$ be such that every positive integer $n$ has a unique representation $n=a_i-a_j$ with $a_i,a_j\in A$. How slowly can the numerator $a_i=a_i^{(n)}$ grow re…

If $A⊆N$ is an infinite Sidon set and $A(x)=|A∩[1,x]|$, must $\liminf_{x→∞} A(x)√{\log x}/√x=0$? Conversely, is there an infinite Sidon set and a constant $c>0$ for which…

Let $A\subseteq \{1,\ldots N\}$ be a set such that there exists at most one $n$ with more than one solution to $n=a+b$ (with $a\leq b\in A$). Estimate the maximal possible size of…

Let $A_1(N)$ be the number of inclusion-maximal Sidon subsets of $\{1,\ldots,N\}$, where a set is Sidon if $a+b=c+d$ for elements of the set implies that the two unordered pairs…

Let $f(N)$ be the size of the largest quasi-Sidon subset $A\subset\{1,\ldots,N\}$, where we say that $A$ is quasi-Sidon if $\lvert A+A\rvert=(1+o(1))\binom{\lvert A\rvert}{2}.$ H…

What is the size of the largest Sidon subset $A\subseteq\{1,2^2,\ldots,N^2\}$? Is it $N^{1-o(1)}$?

For fixed $k≥1$, let $H_k(n)$ be the largest integer such that every $n$-element set of integers in which each integer has at most $k$ representations as a sum of two elements cont…

Let $A\subset \mathbb{R}$ be a set of size $n$ such that every subset $B\subseteq A$ with $\lvert B\rvert =4$ has $\lvert B-B\rvert\geq 11$. Find the best constant $c>0$ such that…

Let $A\subseteq\mathbb{N}$ be finite and Sidon, where Sidon means that for all $a_1,a_2,b_1,b_2\in A$, if $a_1+a_2=b_1+b_2$, then either $a_1=b_1$ and $a_2=b_2$, or $a_1=b_2$ and…

Let $\ell(N)$ be maximal such that in any finite set $A\subset \mathbb{R}$ of size $N$ there exists a Sidon subset $S$ of size $\ell(N)$ (i.e. the only solutions to $a+b=c+d$ in…

Let $F(n)$ be the maximum possible size of a subset $A\subseteq\{1,\ldots,N\}$ such that the products $ab$ are distinct for all $a<b$. Is there a constant $c$ such that…

Let $A=\{1,2,4,8,13,21,31,45,66,81,97,\ldots\}$ be the greedy Sidon sequence: we begin with $1$ and iteratively include the next smallest integer that preserves the Sidon property…

Let $f(N)$ be the maximum size of $A\subseteq \{1,\ldots,N\}$ such that the sums $a+b+c$ with $a,b,c\in A$ are all distinct (aside from the trivial coincidences). Is it true that…

Let $A\subset \mathbb{N}$ be an infinite set such that, for any $n$, there are most $2$ solutions to $a+b=n$ with $a\leq b$. Must…

Does there exist an infinite Sidon set $A⊆N$ such that every sufficiently large integer is the sum of three elements of $A$?

Does there exist a maximal Sidon set $A\subset \{1,\ldots,N\}$ of size $O(N^{1/3})$?

Let $F(N)$ be the size of the largest Sidon subset of $\{1,\ldots,N\}$. Is it true that for every $k\geq 1$ we have $F(N+k)\leq F(N)+1$ for all sufficiently large $N$?