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Register a new userAlmost periodic functions and hyperbolic counting
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Authors
Giacomo Cherubini
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In this paper we prove the existence of asymptotic moments, and an estimate on the tails of the limiting distribution, for a specific class of almost periodic functions. Then we introduce the hyperbolic circle problem, proving an estimate on the asymptotic variance of the remainder that improves a result of Chamizo. Applying the results of the first part we prove the existence of limiting distribution and asymptotic moments for three functions that are integrat
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The goal of this expository article is a fairly self-contained account of some averaging processes of functions along sequences of the form $(alpha^n x)^{}_{ninmathbb{N}}$, where $alpha$ is a fixed real number with $| alpha | > 1$ and $xinmathbb{R}$ is arbitrary. Such sequences appear in a multitude of situations including the spectral theory of inflation systems in aperiodic order. Due to the connection with uniform distribution theory, the results will mostly be metric in nature, which means that they hold for Lebesgue-almost every $xinmathbb{R}$.
The paper develops a method for discrete computational Fourier analysis of functions defined on quasicrystals and other almost periodic sets. A key point is to build the analysis around the emerging theory of quasicrystals and diffraction in the setting on local hulls and dynamical systems. Numerically computed approximations arising in this way are built out of the Fourier module of the quasicrystal in question, and approximate their target functions uniformly on the entire infinite space. The methods are entirely group theoretical, being based on finite groups and their duals, and they are practical and computable. Examples of functions based on the standard Fibonacci quasicrystal serve to illustrate the method (which is applicable to all quasicrystals modeled on the cut and project formalism).
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.
Consider polynomials over ${rm GF}(2)$. We describe efficient algorithms for finding trinomials with large irreducible (and possibly primitive) factors, and give examples of trinomials having a primitive factor of degree $r$ for all Mersenne exponents $r = pm 3 bmod 8$ in the range $5 < r < 10^7$, although there is no irreducible trinomial of degree $r$. We also give trinomials with a primitive factor of degree $r = 2^k$ for $3 le k le 12$. These trinomials enable efficient representations of the finite field ${rm GF}(2^r)$. We show how trinomials with large primitive factors can be used efficiently in applications where primitive trinomials would normally be used.
In this paper, we use the variational approach to investigate recurrent properties of solutions for stochastic partial differential equations, which is in contrast to the previous semigroup framework. Consider stochastic differential equations with monotone coefficients. Firstly, we establish the continuous dependence on initial values and coefficients for solutions. Secondly, we prove the existence of recurrent solutions, which include periodic, almost periodic and almost automorphic solutions. Then we show that these recurrent solutions are globally asymptotically stable in square-mean sense. Finally, for illustration of our results we give two applications, i.e. stochastic reaction diffusion equations and stochastic porous media equations.
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