11 problems

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 $p$ be a sufficiently large prime, and let $\mathcal R_p$ denote the set of nonzero squares in $\mathbb F_p$. For $\lambda\in\mathbb F_p^*$, consider the shifted set…

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…

Let $\mathbb{N}$ be the set of positive integers, and let a finite coloring assign one of finitely many colors to each element of $\mathbb{N}$. A set is monochromatic if all of its…

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…