48 problems

An old problem of mine states as follows: Let $a_1 < a_2 < \cdots < a_k$ be a Sidon sequence. Can one extend it to a larger Sidon sequence … In other words, loosely speaking: can o…

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…

Here is an old conjecture of mine: Let $a_1 < a_2 < \ldots$ be an infinite sequence for which all the triple sums $a_i + a_j + a_k$ are distinct. Is it then true that…

Sidon also asked: Let $A = \{a_1 < a_2 < \ldots\}$ be an infinite sequence for which all the sums $a_i + a_j$ are distinct. Put … Probably there is a Sidon sequence $A$ for which ……

Sidon also asked: Let $1 \leq a_1 < a_2 < \cdots < a_k \leq n$ and assume that $a_i + a_j$ are all distinct. Put $h(n) = \max k$. Determine or estimate $h(n)$ as accurately as poss…

Now I state a few problems of Nathanson and myself : Let $a_1 < a_2 < \cdots$ be an infinite sequence of integers; denote by $f(n)$ the number of solutions of $n = a_i + a_j$. Deno…

Let $f(n)$ be the largest cardinality of a Sidon set contained in $\{1,2,\ldots,n\}$, where a subset $\mathcal{S}$ of the natural numbers is Sidon if all sums $s_1+s_2$ with…

A $K$-fold Sidon set is a subset of $[N]$ containing no non-trivial solutions to equations of the form $\sum_{i=1}^4\alpha_i x_i=0$, where $\sum_{i=1}^4\alpha_i=0$ and the coeffici…

Let a finite Sidon set be a finite set of integers in which every difference between two distinct elements has at most one representation as an ordered difference of elements of th…

Let $A$ be a finite set of integers, and let $B\subseteq A$ and $C\subseteq A$ be, respectively, a $B_2^+[1]$ set and a $B_2^{\times}[1]$ set. Klurman–Pohoata conjecture. In the ab…

Let $R(g,n)$ denote the maximum size of a $B^{\ast}[g]$ set contained in $\{1,\dots,n\}$. A sequence $(S_n)_{n=1}^\infty$ of sets of positive integers becomes uniformly distributed…

Let $\Lambda=\{0,1,2,3\}$, and let its Sidon constant be the least constant controlling the supremum norm of trigonometric polynomials supported on $\Lambda$ by the corresponding s…

Let $h\geq 3$ and let $\mathbf{b}=(b_1,\ldots,b_h)\in\mathbb{Z}^h$ satisfy $\sum_{i=1}^h b_i=0$ and $b_i\neq 0$ for all $i$. Let $A\subseteq\mathbb{N}$, with increasing elements…

Let $A\subset [N]$ be a Sidon set, and for each prime $p$ let $A_p$ denote the set of residue classes modulo $p$ occupied by $A$. Suppose that $|A_p|\leq \alpha p$ for every prime…

Erdős's conjecture. As $n$ tends to infinity,

For an integer $r\geq 1$ and a subset $A\subseteq [n]$, let $c\colon A\to [r]$ be an $r$-coloring. A solution $(x_1,x_2,y_1,y_2)$ to $x_1+x_2=y_1+y_2$ in $A$ is rainbow if the four…

For a positive integer $k$, let $S_k=\{n^k:n\in\mathbb{Z}_{>0}\}$; a set is Sidon if there are no nontrivial equal sums of two of its elements. Lander–Parkin–Selfridge's special-ca…

Additive-or-multiplicative Sidon set conjecture. Any set $A\subset\mathbb{R}$ must contain either a large additive Sidon set or a large multiplicative Sidon set.

Let $S\subseteq \mathbb{F}_2^n$ be a $k$-cover, and suppose that … An AB function is a function $F\colon\mathbb{F}_2^{n/2}\to\mathbb{F}_2^{n/2}$ whose graph is denoted by…

Let $A = \{a_1,\ldots,a_n\}$ be a finite set of positive integers, and let an $m$-mark Golomb ruler be a subset of $A$ with $m$ elements and all pairwise differences distinct. Koml…

Axis-value conjecture. For every $a\in\mathbb{F}_2^n$ and every nonzero $b\in\mathbb{F}_2^n$,

Uniform-exclude-distribution conjecture. If $d_{\mathcal{G}_F}$ is uniform on $\mathcal{Q}(\mathbb{F}_2^n,F)$, then $\mathcal{G}_F$ is maximal.

Maximality conjecture. The graph $\mathcal{G}_F$ is maximal for every APN function $F$.

Single-point modification conjecture. The function $G$ is not APN.