We prove the convergence of moments of the number of directions of affine lattice vectors that fall into a small disc, under natural Diophantine conditions on the shift. Furthermore, we show that the pair correlation function is Poissonian for any irrational shift in dimension 3 and higher, including well-approximable vectors. Convergence in distribution was already proved in the work of Strömbergsson and the second author [The distribution of free path lengths in the periodic Lorentz gas and related lattice point problems. Ann. of Math. (2) 172 (2010), 1949–2033], and the principal step in the extension to convergence of moments is an escape of mass estimate for averages over embedded $\operatorname {SL}(d,\mathbb {R})$-horospheres in the space of affine lattices.

Let $(X,{\mathcal{B}},\unicode[STIX]{x1D707},T)$ be an ergodic measure-preserving system, let $A\in {\mathcal{B}}$ and let $\unicode[STIX]{x1D716}>0$. We study the largeness of sets of the form

$$\begin{eqnarray}S=\{n\in \mathbb{N}:\unicode[STIX]{x1D707}(A\cap T^{-f_{1}(n)}A\cap T^{-f_{2}(n)}A\cap \cdots \cap T^{-f_{k}(n)}A)>\unicode[STIX]{x1D707}(A)^{k+1}-\unicode[STIX]{x1D716}\}\end{eqnarray}$$

$\{f_{1},\ldots ,f_{k}\}$

$f_{i}:\mathbb{N}\rightarrow \mathbb{N}$

$k\leq 3$

$f_{i}(n)=if(n)$

$S$

$f(n)=q(p_{n})$

$q\in \mathbb{Z}[x]$

$q(1)$

$q(-1)=0$

$p_{n}$

$n$

$f$

$T^{q}$

$q\in \mathbb{N}$

$r\in \mathbb{Z}$

$S$

$f(n)=qn+r$

$f_{i}(n)=a_{i}n$

$a_{i}$

$S$

$k\geq 4$

$k=3$

$S$

$f_{i}$

In this paper we establish a uniform distribution result for the time spent by a billiard particle in the unit square having vertices (0,0), (1,0), (1,1), (0,1), with small triangular pockets of size $\epsilon$ removed from its corners.

Constants with formulae of the form treated by D. Bailey, P. Borwein, and S. Plouffe ($b$ formulae to a given base $b$) have interesting computational properties, such as allowing single digits in their base $b$ expansion to be independently computed, and there are hints that they should be normal numbers, i.e., that their base $b$ digits are randomly distributed. We study a formally limited subset of BBP formulae, which we call Machin-type BBP formulae, for which it is relatively easy to determine whether or not a given constant $K$ has a Machin-type BBP formula. In particular, given $b\,\in \,\mathbb{N},\,b\,>\,2,\,b$ not a proper power, a $b$-ary Machin-type BBP arctangent formula for $K$ is a formula of the form $k\,=\,{{\Sigma }_{m}}\,{{a}_{m}}\,\arctan \,(-{{b}^{-m}}),\,{{a}_{m}}\,\in \,\mathbb{Q}$ , while when $b\,=\,2$, we also allow terms of the form ${{a}_{m}}\,\arctan \,(1/1\,-\,{{2}^{m}}))$ . Of particular interest, we show that $\pi$ has no Machin-type BBP arctangent formula when $b\,\ne \,2$. To the best of our knowledge, when there is no Machin-type BBP formula for a constant then no BBP formula of any form is known for that constant.

Let $a$ be a natural number greater than 1. Let ${{f}_{a}}\left( n \right)$ be the order of $a\,\bmod \,n$. Denote by $\omega \left( n \right)$ the number of distinct prime factors of $n$. Assuming a weak form of the generalised Riemann hypothesis, we prove the following conjecture of Erdös and Pomerance:

The number of $n\,\le \,x$ coprime to a satisfying

$$\alpha \le \frac{\omega \left( {{f}_{a}}\left( n \right) \right)-{{\left( \log \,\log \,n \right)}^{2}}/2}{{{\left( \log \,\log \,n \right)}^{3/2}}/\sqrt{3}}\le \beta $$

is asymptotic to $\left( \frac{1}{\sqrt{2\pi }}\int_{\alpha }^{\beta }{{{e}^{-{{t}^{2}}/2}}}dt \right)\frac{x\phi \left( a \right)}{a}$ as $x$ tends to infinity.