750 problems

Let $s\geq 3$ be an integer. An $s$-spanning set of size $2$ in a finite abelian group is a set of two elements whose signed sums of at most $s$ terms cover the group. Largest nonc…

For every integer $n \ge 0$, every real number $K \ge 1$, and every nonempty set $A \subseteq \mathbb{F}_2^n$ satisfying $\lvert A+A\rvert \le K\lvert A\rvert$, there exist a subsp…

For every prime power $q$ and integer $k\ge 0$, let $C_k(q)$ be the least cardinality of a family of nonzero polynomials over $\mathbb{F}_q$ whose associated codimension-$k$ partia…

For every finite nonabelian group $G$, let $\exp(G)$ be the exponent of $G$. Define $s(G)$ to be the least integer $\ell$ such that every sequence over $G$ of length at least…

Fix an integer $t\geq 3$. For a $t$-element set $A\subset\mathbb{Z}$, let $\Delta(A)=\{a_j-a_i:a_i,a_j\in A,\ a_i<a_j\}$ be its set of positive differences. Call $A$ a Golomb ruler…

Let $A$ be a finite subset of the positive integers, and let $n$ be a positive integer. For each positive integer $k$, write $k\boldsymbol{\bullet} A=\{ka:a\in A\}$, and let…

Entropic Marton-type conjecture. There exists $c_2=\exp(O(1/c_1))$ such that, for every $m\in\mathbb{Z}_+$ and every such $X$, there is a subspace $G$ with

Fix $n,m\in\mathbb{N}$ and let … for $A,B\subseteq\mathbb{N}_0$. Let $r,s\in\mathbb{N}$ be multiplicatively independent, and let $A,B\subseteq\mathbb{N}_0$ be $\times r$- and…

Generalized Erdős–Ginzburg–Ziv conjecture. For every finite group $G$,

Let $K$ be an algebraically closed field and let $F$ be an extension field of $K$. Let $S$ be a finite-dimensional $K$-subspace of $F$ such that $K\subset S$. Define the combinator…

A sequence of positive integers is a Sidon sequence if all sums $a+a'$ with $a,a'$ in the sequence and $a\leq a'$ are distinct; it is an asymptotic basis of order $g$ if every suff…

Let $p$ be a prime, and let $s_1(Z_p^3)$ denote the least integer $\ell$ such that every sequence of length $\ell$ in the finite abelian group $Z_p^3$ contains a zero-sum subsequen…

Chung–Goldwasser conjecture. Every measurable 3-sum-free subset $A$ of $[0,1]$ satisfies

Let $f:\mathbb{F}_2^n\to\mathbb{C}$ be a $K$-approximate quadratic, meaning that $f$ has large $U^3$ norm in the sense used in the paper. A quadratic phase is a function of the for…

Let $A$ be an increasing sequence of positive integers, allowing repetitions, and let $A(n)$ denote the number of elements of $A$, counted with multiplicity, that are at most $n$.…

Let $A$ be a strictly increasing sequence of positive integers, and write $A(n):=|A\cap[n]|$, where $[n]:=\{1,2,\dots,n\}$. Thus $A$ has asymptotic density at least $C\sqrt n$ when…

Let $q$ be prime, let $g_1,\ldots,g_k$ be independent uniformly random generators of the cyclic group ${\mathbb Z}_q$, and let $Diam(q,k)$ denote the diameter of the resulting dire…

Let $j$ and $k$ be nonnegative integers with $k$ even. Prescribed-defect existence conjecture. There exists a positive integer $n$ and a set $S\subset\{0,1,\dots,n-1\}$ such that…

For $n\ge 1$, let $S$ range over subsets of $\{0,1,\dots,n-1\}$. For nonnegative integers $j$ and $k$, define … assuming the limit exists. Since $S-S$ is symmetric about $0$, only…

Limiting-proportion conjecture. The three limiting proportions all exist and are positive.

The short square-set energy conjecture. One should have

For a finite set $E$ of squares in the short interval … let $r_{E+E}(m)$ count representations of $m$ as a sum of two elements of $E$. The local energy conjecture. There exists…

For a set $E\subset\{1^2,\ldots,N^2\}$ of squares, let $r_{E+E}(m)$ count representations of $m$ as a sum of two elements of $E$. The logarithmic energy conjecture. There exists a…

An affine cube of dimension $d$ in $\mathbb Z$ is a set … where $b_0,\ldots,b_d$ are nonzero integers. Solymosi's conjecture. There exists an integer $d>0$ such that no affine cube…

For a finite set $E$ of squares, write $E+E=\{x+y:x,y\in E\}$. Ruzsa's conjecture. For every $\epsilon>0$, … The source states that Chang's conjecture implies this one, while the B…