The Erdős-Szüsz-Turán distribution for equivariant processes (Q2314865)

The paper is a contribution to the probabilistic Diophantine approximation. Let \(A>0,c>0, N>0\). For each \(\alpha \in [0,1]\), denote by \(\mathrm{EST}(A,c,N)(\alpha)\) the number of solutions \(p/q\in \mathbb{Q}\) with \(\mathrm{gcd}(p,q)=1\) of \[ \left| \alpha - \frac{p}{q}\right| \leq \frac{A}{q^2}, \quad N \leq q \leq cN. \] Letting \(\alpha\) be uniformly distributed on \([0,1]\), we obtain an integer-valued random variable, denoted by \(\mathrm{EST}(A,c,N)\), whose law is \[ \mathbb{P}(\mathrm{EST}(A,c,N)=k)=Leb(\alpha\in [0,1]: \mathrm{EST}(A,c,N)(\alpha)=k). \] \textit{P. Erdős} et al. [Colloq. Math. 6, 119--126 (1958; Zbl 0087.04305)] proved that if \(A\leq \frac{c}{1+c^2}\), then \[ \lim_{N\to \infty} \mathbb{P}( \mathrm{EST}(A,c,N)>0 )= \frac{12 A}{\pi^2} \log c. \] The condition \(A\leq \frac{c}{1+c^2}\) was later loosened and removed by \textit{H. Kesten} [Trans. Am. Math. Soc. 103, 189--217 (1962; Zbl 0105.03805)] and by \textit{H. Kesten} and \textit{V. T. Sós} [Acta Arith. 12, 183--192 (1966; Zbl 0144.28604)] respectively. Kesten [loc. cit.] also studied the sequence of random variables \(\mathrm{K}(A,N)\) which are defined similarly by replacing the above inequalities by \[ \left| q\alpha - p \right| \leq \frac{A}{N}, \quad 1 \leq q \leq N. \] The first aim of the present paper is to find the limiting distributions of the random variables \(\mathrm{EST}(A,c,N)\) and \(\mathrm{K}(A,N)\). It is shown that the problems can be easily translated into problems on dynamics on the space of unimodular lattices and then some classical equidistribution results can be applied to reduce the mentioned questions to computing the probability of a random unimodular lattice intersecting a certain fixed region. Precisely, let \(\mu_2\) denote the Haar probability measure on the space \(X_2:=\mathrm{SL}(2, \mathbb{R})/\mathrm{SL}(2, \mathbb{Z})\) of unimodular lattices and for any \(\Lambda \in X_2\), let \(\Lambda_\mathrm{prim}\) be the set of primitive vectors in \(\Lambda\). Using Zagier's equidistribution theorem [\textit{D. Zagier}, in: Automorphic forms, representation theory and arithmetic, Pap. Colloq., Bombay 1979, 275--301 (1981; Zbl 0484.10019)], the authors prove that \[ \lim_{N\to\infty} \mathbb{P}(\mathrm{EST}(A,c,N)=k)= \mu_2(\Lambda\in X_2: \sharp (\Lambda_\mathrm{prim} \cap H_{A,c})=k), \] and \[ \lim_{N\to\infty} \mathbb{P}(\mathrm{K}(A,N)=k)= \mu_2(\Lambda\in X_2: \sharp (\Lambda_\mathrm{prim} \cap R_{A})=k), \] with \[ H_{A,c}:=\{(x,y)\in \mathbb{R}^2: xy\leq A, 1\leq y \leq c\}, \] and \[ R_A:=\{(x,y)\in \mathbb{R}^2: |x|\leq A, 0\leq y \leq 1\}. \] The authors also generalize their results to higher dimensional cases and to the approximation on curves. Then instead of Zagier's equidistribution theorem, the equidistribution theorems by \textit{C. A. Rogers} [Acta Math. 94, 249--287 (1955; Zbl 0065.28201)] and \textit{N. A. Shah} [Invent. Math. 177, No. 3, 509--532 (2009; Zbl 1210.11083)] are used. In fact, the authors can generalize the problem to the setting of equivariant measure-valued processes in \(\mathbb{R}^n\). Then by finding appropriate equidistribution results, one can solve many Erdős-Szüsz-Turán- and Kesten-type problems. As an application, the authors give the corresponding results for translation surfaces.