14 problems

Let $A$ be a finite set of integers, and let $B\subseteq A$ and $C\subseteq A$ be, respectively, a $B_2^+[1]$ set and a $B_2^{\times}[1]$ set. Klurman–Pohoata conjecture. In the ab…

Let $d\geq 2$ be fixed. Let $q\equiv 1\pmod d$ be a sufficiently large prime power, and let $G$ be the multiplicative subgroup of $\mathbb F_q$ of index $d$. For each…

Let $n\ge 3$ and let $A\subseteq \mathbb R$ be a Borel set. Write $\dim_H A$ for the Hausdorff dimension of $A$. Shifted-product nonempty-interior conjecture. For every $n\ge 3$, t…

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

Let $[n]_p$ be the binomial random subset of $[n]$, obtained by retaining each integer independently with probability $p$, and call $[n]_p$ product-Schur if every $2$-colouring of…

Let $R$ be obtained from the set $\{x^k+1:x\in\mathbb{N}\}$ by changing $o(X^{1/k})$ elements up to $X$, where $k\geq 2$. A set $S\subset\mathbb{N}$ is multiplicatively irreducible…

Multiplicative bounded-product conjecture. There exist an infinite set $B\subset\mathbb{N}$ and $t\in\mathbb{N}$ such that

Sárközy's conjecture. If $q$ is sufficiently large, then for every $\lambda\in\mathbb{F}_q^*$, the shifted subgroup $(S_d-\lambda)\setminus\{0\}$ has no nontrivial multiplicative d…

Let $X_k$ be the underlying set of the Gower subspaces, and let $X_k=\bigcup_{i=1}^r C_i$ be a finite partition. A Gower sum subspace is generated by a subset…

Multiplicative growth conjecture. For every integer $m$, there is an integer $n=n(m)$ such that $|Q^n| \gg |Q|^m$. This hypothesis concerns the multiplicative expansion of combinat…

Let $\delta_{\text{min}}(\{1,2,3\})$ denote the sharp minimum lower asymptotic density of a multiplicatively $\{1,2,3\}$-syndetic set, as defined by the Graham–Spencer–Witsenhausen…

A subset $S\subseteq\mathbb{N}$ is syndetic if there is an integer $M\geq 1$ such that every interval of $M$ consecutive positive integers meets $S$. A geometric progression in $S$…

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 $q$ be a large prime, and let $P_1$ be the set of all primes less than $q$, regarded as a subset of the multiplicative group $\left(\mathbb{Z}/q\mathbb{Z}\right)^{\times}$. For…