22 problems

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 $p$ 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…

Erdős–Szemerédi conjecture. For any $g\in\mathbb{N}$, any $\varepsilon>0$ and any set $A\subseteq\mathbb{C}$, one should have

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…

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

Elekes–Ruzsa conjecture. If

Katz–Tao's discretized ring conjecture. There exists an absolute constant $c_1$ such that, if $\varepsilon>0$ is sufficiently small and $C$ is sufficiently large, then for all suff…

Klurman–Pohoata conjecture. There exists a constant $c>0$ such that, for every finite $A\subset\mathbb R$,

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 subset of $\mathbb{R}$. For a finite set $X$, write … where $E_k^+(X)=\sum_x r_{X-X}^k(x)$ and $E_k^\times(X)=\sum_x r_{X/X}^k(x)$. Let $\eta\leq 2/3$ be given.…

Let $A\subset\mathbb{F}_p$, and let $\epsilon>0$. Iosevich's conjecture. If … then … The conjecture concerns simultaneous additive-multiplicative expansion in prime fields and is s…

Two-line intersection conjecture. For all $\delta>0$, there exists $\epsilon=\epsilon(\delta)>0$ for which the displayed bound holds for every such $P$, $l_1$, and $l_2$. This conj…

Let $A\subset\mathbb{R}$ be finite, and define … with the convention $0^{-1}=0$. Also let $AA=\{ab:a,b\in A\}$. Difference-product and difference-quotient conjecture. For every fix…

Balog's conjecture. For every such set $A$,

Let $A$ be a set of scalars with at least three elements, and put $D=A-A$. For a positive integer $m$, write $D^m$ for the $m$-fold product set of $D$. Balog–Roche-Newton–Zhelezov…

Let $X\subset\mathbb{R}$ be a finite real set, and let $XX=\{xy:x,y\in X\}$. Let $A\subset\mathbb{R}$ be a set with difference set $A-A=\{a-a':a,a'\in A\}$. Small-product-set diffe…

Let $A\subset\mathbb{R}$ be a finite set, and for a positive integer $k$ write … Here $X\gg_{k,l}Y$ means that $X\geq c_{k,l}Y$ for a positive constant depending only on $k$ and…

Let $A$ be a finite subset of real numbers. Define the sumset and productset by … Sum-product conjecture. For arbitrarily small positive $\epsilon$, … This is a central conjecture…

Let $\textsc{Squares}$ denote the set of perfect squares in $\mathbb Z$. Finite-translate perfect-squares conjecture. There exists a positive integer $k$ such that there are no set…