7 problems

Bloom–Walker conjecture. Under this hypothesis, the sequence $(a_n)$ has metric Poissonian pair correlation; that is, for every $s>0$,

Near-quadratic energy conjecture. If $A\subset\mathbb{R}$ is convex, then

Convex-sequence additive energy conjecture. For every $varepsilon>0$, one has

Incidence-energy conjecture. The second term should improve to

Let $A,B\subseteq\mathbb{Z}$ be sets of size $N$ satisfying … Write $A'=\{a'_1<\cdots<a'_k\}$ and $B=\{b_1<\cdots<b_n\}$. Generalized indexed-energy conjecture. There exist constan…

Optimal carries conjecture. For every $\epsilon>0$, one should have

Let $N>0$ be a large parameter, with constants independent of $N$. Let $B,C\subseteq F_3^N$ satisfy $|B|,|C|\leq N^{100}$. Suppose that the number of quadruples…