Expansion, divisibility and parity

Let $\mathbf{P}\subset [H_0,H]$ be a set of primes, where $\log H_0 \geq (\log H)^{2/3 + \epsilon}$\mathrm{.}Let$\mathscr{L} = \sum_{p \in $\mathbf{P}$} 1/p$. Let $N$ be such that $\log H\le (\log N{)}^{1\mathrm{/}2-ϵ}$. We show there exists a subset $\mathcal{X}\subset (N,2N]$ of density close to $1$ such that all the eigenvalues of the linear operator $$(A_{\mathrm{\mid }\mathcal{X}}f)(n)=\sum _{\begin{array}{c}p\in \mathbf{P}:p\mathrm{\mid }n\\ n,n\pm p\in \mathcal{X}\end{array}}f(n\pm p)-\sum _{\begin{array}{c}p\in \mathbf{P}\\ n,n\pm p\in \mathcal{X}\end{array}}\frac{f(n\pm p)}{p}$$ are $O(\sqrt{\mathcal{L}})$. This bound is optimal up to a constant factor. In other words, we prove that a graph describing divisibility by primes is a strong local expander almost everywhere, and indeed within a constant factor of being “locally Ramanujan” (a.e.). Specializing to $f(n)=\lambda (n)$ with $\lambda (n)$ the Liouville function, and using an estimate by Matom"aki, Radziwi{\l}{\l} and Tao on the average of $\lambda (n)$ in short intervals, we derive that $$\frac{1}{\log x}\sum _{n\le x}\frac{\lambda (n)\lambda (n+1)}{n}=O\left(\frac{1}{\sqrt{\log \log x}}\right),$$ improving on a result of Tao’s. We also prove that $\sum_{N<n\leq 2 N} \lambda(n) \lambda(n+1)=o(N)$ at almost all scales with a similar error term, improving on a result by Tao and Ter"av"ainen. (Tao and Tao-Ter"av"ainen followed a different approach, based on entropy, not expansion; significantly, we can take a much larger value of $H$, and thus consider many more primes.) We can also prove sharper results with ease. For instance: let $S_{N,k}$ the set of all $N<n\le 2N$ such that $\mathrm{\Omega }(n)=k$. Then, for any fixed value of $k$ with $k=\log \log N+O(\sqrt{\log \log N})$ (that is, any “popular” value of $k$) the average of $\lambda (n+1)$ over $S_{N,k}$ is $o(1)$ at almost all scales.