Best possible rates of distribution of dense lattice orbits in homogeneous spaces

The present paper establishes upper and lower bounds on the speed of approximation in a wide range of natural Diophantine approximation problems. The upper and lower bounds coincide in many cases, giving rise to optimal results in Diophantine approximation which were inaccessible previously. Our approach proceeds by establishing, more generally, upper and lower bounds for the rate of distribution of dense orbits of a lattice subgroup $\Gamma$ in a connected Lie (or algebraic) group $G$, acting on suitable homogeneous spaces $G/H$. The upper bound is derived using a quantitative duality principle for homogeneous spaces, reducing it to a rate of convergence in the mean ergodic theorem for a family of averaging operators supported on $H$ and acting on $G/\Gamma$. In particular, the quality of the upper bound on the rate of distribution we obtain is determined explicitly by the spectrum of $H$ in the automorphic representation on $L^2(\Gamma\setminus G)$. We show that the rate is best possible when the representation in question is tempered, and show that the latter condition holds in a wide range of examples

Exponential prefixed polynomial equations

A prefixed polynomial equation is an equation of the form $P(t_1,\ldots,t_n) = 0$, where $P$ is a polynomial whose variables $t_1,\ldots,t_n$ range over the natural numbers, preceded by quantifiers over some, or all, of its variables. Here, we consider exponential prefixed polynomial equations (EPPEs), where variables can also occur as exponents. We obtain a relatively concise EPPE equivalent to the combinatorial principle of the Paris-Harrington theorem for pairs (which is independent of primitive recursive arithmetic), as well as an EPPE equivalent to Goodstein's theorem (which is independent of Peano arithmetic). Some new devices are used in addition to known methods for the elimination of bounded universal quantifiers for Diophantine predicates

Ultrametric Logarithm Laws, II

We prove positive characteristic versions of the logarithm laws of Sullivan and Kleinbock-Margulis and obtain related results in Metric Diophantine Approximation.Comment: submitted to Montasefte Fur Mathemati