13 problems

Let $n,k\in\mathbb{N}$, let $\lambda>0$, and let $\delta>0$. For a set $A\subseteq\{1,\ldots,n\}$, its doubling factor is measured by $|A+A|/|A|$. The construction underlying the c…

Small-doubling abelianity conjecture. If

Three-set small-doubling conjecture. If $|S^2|\leq 3k-4$, then $\langle S\rangle$ is abelian.

Two-set small-doubling conjecture. If $|S^2|\leq 3k-3$, then $\langle S\rangle$ is abelian.

Let $G$ be a finite abelian group of exponent $t$, and let $A\subseteq G$ satisfy … A subgroup containing $A$ is a subgroup whose size bounds the ambient structured set containing…

Let $G$ be an abelian group of exponent $r$, and let $A\subseteq G$ have size $n$ and satisfy … A subgroup containing $A$ is a subgroup of $G$ that contains $A$. Ruzsa's conjecture…

Let $n$ be a positive integer, let $\mathbb{Z}_n$ be the cyclic group of order $n$, and let $A\subseteq\mathbb{Z}_n$. For a subset $S\subseteq\mathbb{Z}_n$, write $2S=S+S$. Small-d…

Let $[n]=\{1,\ldots,n\}$, and let $s$ and $K$ be parameters. A set $J\subset[n]$ has doubling constant at most $K$ when $|J+J|\leq K|J|$. Alon–Balogh–Morris–Samotij conjecture. For…

Low-dimensional subset conjecture. There is a subset $X \subseteq A$ such that

Let $A,B\subseteq\mathbb{Z}$ be sets of size $N$ satisfying … Write $A'=\{a'_1<\cdots<a'_k\}$ and $B=\{b_1<\cdots<b_n\}$. Generalized indexed-energy conjecture. There exist constan…

Let $G$ be a torsion-free group and let $X$ be a subset of $G$. Write $2X=X+X$ in additive notation, and let $\langle X\rangle$ denote the subgroup generated by $X$. Freiman's conj…

Almost-homomorphism formulation of the Polynomial Freiman-Ruzsa conjecture. Is it true that $f=\widetilde f+g$, where $\widetilde f:\mathbb{F}_2^n\rightarrow\mathbb{F}_2^{n'}$ is l…

Let $F(K)$ and $G(K)$ be the functions introduced earlier in the paper for measuring the linear structure of finite binary sets with doubling parameter $K$. The piecewise-linearity…