11 problems

The short square-set energy conjecture. One should have

For a finite set $E$ of squares in the short interval … let $r_{E+E}(m)$ count representations of $m$ as a sum of two elements of $E$. The local energy conjecture. There exists…

The short interval conjecture. For every $\alpha<1$, every such trigonometric polynomial satisfies

For a set $E\subset\{1^2,\ldots,N^2\}$ of squares, let $r_{E+E}(m)$ count representations of $m$ as a sum of two elements of $E$. The logarithmic energy conjecture. There exists a…

For integers $a$ and $b$, consider positive solutions of $x^2\equiv a\pmod b$. The congruence-spacing conjecture. There exists a constant $N$ such that, for every $a$ and $b$, ther…

An affine cube of dimension $d$ in $\mathbb Z$ is a set … where $b_0,\ldots,b_d$ are nonzero integers. Solymosi's conjecture. There exists an integer $d>0$ such that no affine cube…

For a finite set $E$ of squares, write $E+E=\{x+y:x,y\in E\}$. Ruzsa's conjecture. For every $\epsilon>0$, … The source states that Chang's conjecture implies this one, while the B…

Let $R$ be obtained from the set of positive squares by deleting some of its elements up to $X$ and adding some positive integers, with the total number of changed elements being…

Let $B\in\mathbb{N}_{\ge 2}$ and let $D\subseteq\{0,1,\dots,B-1\}$ satisfy $|D|\ge 2$. Let $\alpha_B^D(n)$ be the number of integers $B^{n-\frac12}\le x<B^{n+\frac12}$ such that th…

For a base $B\ge 2$ and a digit set $D\subseteq\{0,1,\dots,B-1\}$ with $|D|\ge 2$, let $\omega_B^D(n)$ denote the number of $n$-digit suffixes in base $B$ from $D$ that occur as th…

Let $\langle a_1',\dots,a_l'\rangle$ be a solution of length $l\ge 2$, where a solution is a digit sequence satisfying the paper's square-digit condition. Prefix-closure conjecture…