51 problems

Balog's conjecture. One has

Let a Borel subgroup or Borel subring of the reals mean a subgroup or subring of $\mathbb{R}$ that is Borel as a subset, and let Hausdorff dimension refer to its dimension as a sub…

Weak polynomial Freiman–Ruzsa conjecture. There is an absolute constant $c>0$ such that one can find $A_1\subseteq A$ with

Balog–Roche-Newton–Zhelezov conjecture. For any $s>0$, there exists $m=m(s)$ such that

For $0<\sigma<1$, define a $(\delta,\sigma)_1$-set to be a $\delta$-separated subset $A\subset[1,2]$ satisfying $|A|\sim\delta^{-\sigma}$ and $|A\cap I|\lesssim |I|^{\sigma}|A|$ fo…

Let $A$ be a finite set of real numbers, and define … Here, $X\ll Y$ means that $X$ is bounded above by a constant multiple of $Y$. Roche-Newton–Zhelezov conjecture. … The conjectu…

Erdős's sum-product conjecture. One might conjecture that

Let $\mathbb{A}$ be an annulus in $\mathbb{R}$, and let $0<\delta\ll1$. Let $A\subset\mathbb{A}$ be a $(\delta,1/2)_1$ set with measure $\approx\delta^{1/2}$. Discretized Erdős rin…

Let $X$ and $X'$ be independent and identically distributed, discrete, real-valued random variables with finite entropy $H(X)<\infty$. Entropic Solymosi inequality. One has … where…

Let $X$ and $X'$ be independent and identically distributed, discrete, real-valued random variables with finite entropy $H(X)<\infty$. Entropic sum-product conjecture with optimal…

Let $X$ be a discrete real-valued random variable with finite entropy $H(X)<\infty$, and let $X'$ be an independent copy of $X$. Goh's entropic sum-product conjecture. There exists…

Let $A\textbackslash subset\textbackslash mathbb\textbackslash Z$ be a finite set, and write … Here $o(1)$ denotes a quantity tending to $0$ as…

Folklore conjecture. The exponent in this inequality can be taken to be arbitrarily close to $2$.

Let $A\subset\mathbb{R}$ be a finite set. Discretised sum-product conjecture. For every $\varepsilon>0$, there exists $c>0$ such that … This is the discrete form of the sum-product…

Let $p$ be prime and let $A\subset\mathbb{F}_p$ satisfy $|A|=o(p)$ and $|A|\geq p^{1-c}$ for some sufficiently small constant $c>0$. Sharp sum-product conjecture. Then … This conje…

Let $A\subset \mathbb{R}$ be a finite set. The set $A+AA=\\{a+bc:a,b,c\in A\\}$ combines addition and multiplication. Folklore sum-product conjecture. One should have … This is a f…

Let $A\subset \mathbb{Z}$ be a finite set of integers and let $m\geqslant 1$. Write $mA=A+\cdots+A$ and $A^{(m)}=A\cdots A$. Iterated sum-product conjecture. One should have … Bour…

Erdős's sum-product conjecture. For every $\epsilon>0$ and all sufficiently large finite sets $X\subseteq\mathbf{N}$,

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.

Write $\mathbb{P}$ for the set of primes. Prime-minus-one sum-product conjecture. For every $a\in\mathbb{N}$ there exist $x,y\in\mathbb{N}$ with $x>y>a$ such that … This asks for s…

Let $A\subseteq\mathbb{N}$ have positive density, where its density is … whenever this limit exists. Moreira's conjecture. There exist $x,y\in\mathbb{N}\backslash\{1\}$ such that ……

Let $k\in\mathbb{N}$ and let a finite coloring of $\mathbb{N}$ be given. Generalized Hindman's conjecture. There exist infinitely many $x_1<\ldots<x_k\in\mathbb{N}$ such that all f…

Near-quadratic expansion conjecture. For every $\varepsilon>0$,

Let $p$ be prime and let $X$ be a random variable taking values in the finite field $\mathbb{F}_p$. Suppose that the quantitative condition described immediately beforehand rules o…

Let $A$ be a finite subset of the integers, and write $A+A=\{a_1+a_2:a_1,a_2\in A\}$ and $A\cdot A=\{a_1a_2:a_1,a_2\in A\}$. Erdős–Szeremédi conjecture. For every $0<\epsilon<1$, t…