A conjecture of Cilleruelo and Cordoba and divisors in a short interval

Abstract: Let $E(A)=\#\{(a_1,a_2,a_3,a_4)\in A^4: a_1+a_2=a_3+a_4\}$ denote the additive energy of a set $A\subset \N$, and let $\mathbb{T}=\R/\Z$ and $\|f\|_4=\left(\int_{\mathbb{T}}|f(t)|^4dt\right)^{1/4}$. It is well-known that $$ E(\{n^2: n\leq N\})=\left\|\sum_{n\leq N}e^{2\pi in^2x}\right\|_4^4 \asymp N^2\log N, $$ while we trivially have $E(A)\geq |A|^2$. In 1992, J. Cilleruelo and A. Cordoba proved that $E(\{n^2: N\leq n\leq N+N^{\gamma}\})\asymp N^{2\gamma}$ for any $\gamma\in (0,1)$, and conjectured a much more general bound (again, for any $\gamma\in(0,1)$) $$ \left\|\sum_{N\leq n\leq N+N^{\gamma}}a_ne^{2\pi in^2x}\right\|_4\leq C(\gamma)\left(\sum_{N\leq n\leq N+N^{\gamma}}|a_n|^2\right)^{1/2}. $$ While this bound is easy to prove for $\gamma\leq 1/2$, it seems to be open for any $\gamma>1/2$. We prove this for all $\gamma<\frac{\sqrt5-1}{2}=0.618...$ and present a connection between this problem and a conjecture of I. Ruzsa: for any $\epsilon>0$ there exists $C(\epsilon)>0$ such that any positive integer $N$ has at most $C(\epsilon)$ divisors in the interval $[N^{1/2}, N^{1/2}+N^{1/2-\epsilon}]$.

number theory

Audience: researchers in the discipline

Combinatorial and additive number theory (CANT 2022)