On the number of gaps of sequences with Poissonian pair correlations (Q1981679)

Suppose that the sequence \((x_n)\) in \((0;1)\) has Poissonian pair correlations, i.e., for any real \(s>0\) \[\lim_{N\to\infty} \frac{\sum\limits_{1\leq m\neq n\leq N}\chi_{[0,\frac{s}{N})}||x_n-x_m||}{N}=2s.\] Let \(g(n)\) be a number of different gaps between neighboring elements of \(\{x_1,\ldots,x_n\}\). For \(i=1,\ldots,g(n)\) let \(\varphi_{n,i}\) be the multiplicities of the neighboring gap lengths. Then \(\max_{i\leq g(n)} \varphi_{n,i}=o(n)\). Furthermore, for any function \(f\) with \(\lim_{n\to\infty} f(n)=\infty\) there exists a sequence \((x_n)\) with Poissonian pair correlations such that \(g(n)\leq f(n)\) for all sufficiently large \(n\).