115 problems

Let $\mathscr P$ denote the set of all primes. For sets $S,T$ of positive integers, define $S\sim T$ to mean that their symmetric difference is finite. Ostmann's inverse Goldbach c…

Marton's conjecture. $A$ is $\exp(O(\log K))$-covered by a subgroup $M$ of size at most $\exp(O(\log K))|A|$.

Sharp sumset lower-bound conjecture. The inequality

Let $A\subset\mathbb{F}_{2}^{n}$ be a set with $|A+A|\leqslant K|A|$. Polynomial Freiman-Ruzsa conjecture in characteristic two. Then $A$ is covered by at most $2K^{C}$ cosets of s…

The asymptotic sum-dilate conjecture. $$

Freiman–Lev conjecture. One has

Let $A$ be a set of $k\geq 4$ positive integers, and let $h$ be an integer with $3\leq h\leq k-1$. The restricted signed $h$-fold sumset is denoted by $h^{\wedge}_{\pm}A$. Bhanja–K…

Let $k\in\mathbb{N}$ and let $\epsilon$ be positive. For finite sets $A,B\subset\mathbb{Z}^k$, say that $B$ is not covered by $n$ parallel hyperplanes when no collection of $n$ par…

Let $G$ be an abelian group, let $p$ be the smallest prime divisor of the order of $G$, and let $A$ be an $m$-subset of $G$. Assume that $m$ and $h$ are positive integers satisfyin…

Optimal sumset bounds conjecture. Every such set satisfies

Ruzsa's conjecture. If there exists a constant $K$ such that

Let $k,T,d\bin \mathbb{N}$, and let $c,b$ satisfy … where … Here $\operatorname{vol}(k,T,d)$ is the maximum volume of a set of integers of cardinality $k$, doubling $T$, and additi…

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.

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…

Structural conjecture. There exists a $K\in\mathbb N$ such that for every finite set $A\subseteq\mathbb N$ with $|A|>K$, if

Freiman's conjecture. There exists a natural number $K$ such that for any finite set of natural numbers $A$ with $|A|=k>K$ and

The sumset obstruction conjecture. If, for some $\kappa$, one has

For $cdelta cin (0,1)$ and $Ncinmathbb{N}$, let $cphi_{cdelta}(N)$ be the largest integer such that every set $Acsubseteq [N]$ with $|A|cgeq cdelta N$ contains a sumset $B+C$, wher…

Let $d\geq 2$ be fixed, let $q$ be a prime power with $q\equiv 1\pmod d$, and let $G\subseteq {\mathbb F}_q^*$ be the multiplicative subgroup of index $d$. Generalized Sárközy conj…

Let $A,B \subseteq ewline \mathbb Z$ be nonempty, and let $\mathbf F=(F_n)_{n\in\mathbb N}$ be a Følner sequence of finite subsets of $\mathbb Z$, meaning that … for every…

Let $t$ be a positive integer, let $G$ be an abelian group, and let $A,B\subseteq G$ be finite subsets with $|A|,|B|\geq t$. Write $A+_iB$ for the set of elements of $G$ having at…

Generalized sumset pattern conjecture. There exists an infinite set $B\subset\mathbb N$ such that, for every $i=1,\ldots,k$,

$3k-4$ conjecture. If