We study the variance of the random variable that counts the number of lattice points in some shells generated by a special class of finite type domains in $mathbb R^d$. The proof relies on estimates of the Fourier transform of indicator functions of convex domains.
The hyperbolic lattice point problem asks to estimate the size of the orbit $Gamma z$ inside a hyperbolic disk of radius $cosh^{-1}(X/2)$ for $Gamma$ a discrete subgroup of $hbox{PSL}_2(R)$. Selberg proved the estimate $O(X^{2/3})$ for the error term for cofinite or cocompact groups. This has not been improved for any group and any center. In this paper local averaging over the center is investigated for $hbox{PSL}_2(Z)$. The result is that the error term can be improved to $O(X^{7/12+epsilon})$. The proof uses surprisingly strong input e.g. results on the quantum ergodicity of Maa{ss} cusp forms and estimates on spectral exponential sums. We also prove omega results for this averaging, consistent with the conjectural best error bound $O(X^{1/2+epsilon})$. In the appendix the relevant exponential sum over the spectral parameters is investigated.
In this short note, we reformulate the task of calculating the pair correlation statistics of a Kronecker sequence as a lattice point counting problem. This can be done analogously to the lattice based approach which was used to (re-)prove the famous three gap property for Kronecker sequences. We show that recently developed lattice point counting techniques can then be applied to derive that a certain class of Kronecker sequences have $beta$-pair correlations for all $0 < beta < 1$.
We give two variations on a result of Wilkies on unary functions defianble in $mathbb{R}_{an,exp}$ that take integer values at positive integers. Provided that the functions grows slower than the function $2^x$, Wilkie showed that is must be eventually equal to a polynomial. We show the same conclusion under a stronger growth condition but only assuming that the function takes values sufficiently close to a integers at positive integers. In a different variation we show that it suffices to assume that the function takes integer values on a sufficiently dense subset of the positive integers(for instance primes), again under a stronger growth bound than that in Wilkies result.
In this paper, we derive some identities involving special numbers and moments of random variables by using the generating functions of the moments of certain random variables. Here the related special numbers are Stirling numbers of the first and second kinds, degenerate Stirling numbers of the first and second kinds, derangement numbers, higher-order Bernoulli numbers and Bernoulli numbers of the second kind.
This survey paper examines the effective model theory obtained with the BSS model of real number computation. It treats the following topics: computable ordinals, satisfaction of computable infinitary formulas, forcing as a construction technique, effective categoricity, effective topology, and relations with other models for the effective theory of uncountable structures.