
Dimitrios ChatzakosUniversity of Patras | UP · Department of Mathematics
Dimitrios Chatzakos
PhD
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16
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Introduction
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September 2012 - September 2016
Publications
Publications (16)
We address the prime geodesic theorem in arithmetic progressions and resolve conjectures of Golovchanskiĭ–Smotrov (1999). In particular, we prove that the traces of closed geodesics on the modular surface do not equidistribute in the reduced residue classes of a given modulus.
We address the prime geodesic theorem in arithmetic progressions, and resolve conjectures of Golovchanski\u{\i}-Smotrov (1999). In particular, we prove that the traces of closed geodesics on the modular surface do not equidistribute in the reduced residue classes of a given modulus.
The remainder EΓ(X)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_\Gamma (X)$$\end{document} in the Prime Geodesic Theorem for the Picard group Γ=PSL(2,Z[i])\documen...
We study the fine distribution of lattice points lying on expanding circles in the hyperbolic plane $\mathbb{H}$. The angles of lattice points arising from the orbit of the modular group $PSL_{2}(\mathbb{Z})$, and lying on hyperbolic circles, are shown to be equidistributed for generic radii. However, the angles fail to equidistribute on a thin set...
We study a refinement of the quantum unique ergodicity conjecture for shrinking balls on arithmetic hyperbolic manifolds, with a focus on dimensions $ 2 $ and $ 3 $. For the Eisenstein series for the modular surface $\mathrm{PSL}_2( {\mathbb Z}) \backslash \mathbb{H}^2$ we prove failure of quantum unique ergodicity close to the Planck-scale and an...
Let X(D,1)=Γ(D,1)\H denote the Shimura curve of level N=1 arising from an indefinite quaternion algebra of fixed discriminant D. We study the discrete average of the error term in the hyperbolic circle problem over Heegner points of discriminant d<0 on X(D,1) as d→−∞. We prove that if |d| is sufficiently large compared to the radius r≈logX of the...
The remainder $E_\Gamma(X)$ in the Prime Geodesic Theorem for the Picard group $\Gamma = \mathrm{PSL}(2,\mathbb{Z}[i])$ is known to be bounded by $O(X^{3/2+\epsilon})$ under the assumption of the Lindel\"of hypothesis for quadratic Dirichlet $L$-functions over Gaussian integers. By studying the second moment of $E_\Gamma(X)$, we show that on averag...
Let $X(D,1) =\Gamma(D,1) \backslash \mathbb{H}$ denote the Shimura curve of level $N=1$ arising from an indefinite quaternion algebra of fixed discriminant $D$. We study the discrete average of the error term in the hyperbolic circle problem over Heegner points of discriminant $d <0$ on $X(D,1)$ as $d \to -\infty$. We prove that if $|d|$ is suffici...
For $\Gamma$ a cofinite Kleinian group acting on $\mathbb{H}^3$, we study the Prime Geodesic Theorem on $M=\Gamma \backslash \mathbb{H}^3$, which asks about the asymptotic behaviour of lengths of primitive closed geodesics (prime geodesics) on $M$. Let $E_{\Gamma}(X)$ be the error in the counting of prime geodesics with length at most $\log X$. For...
For $\Gamma$ a cofinite Kleinian group acting on $\mathbb{H}^3$, we study the Prime Geodesic Theorem on $M=\Gamma \backslash \mathbb{H}^3$, which asks about the asymptotic behaviour of lengths of primitive closed geodesics (prime geodesics) on $M$. Let $E_{\Gamma}(X)$ be the error in the counting of prime geodesics with length at most $\log X$. For...
In this thesis we investigate two different lattice point problems in the hyperbolic plane, the classical hyperbolic lattice point problem and the hyperbolic lattice point problem in conjugacy classes. In order to study these problems we use tools from the harmonic analysis on the hyperbolic plane H.
For \Gamma a cofinite Fuchsian group, we study the lattice point problem in conjugacy classes on the Riemann surface \Gamma\backslash\mathbb H . Let \mathcal{H} be a hyperbolic conjugacy class in \Gamma and \ell the \mathcal{H} -invariant closed geodesic on the surface. The main asymptotic for the counting function of the orbit \mathcal{H} \cdot z...
For $\Gamma$ a Fuchsian group of finite covolume, we study the lattice point problem in conjugacy classes on the Riemann surface $\Gamma \backslash \mathbb{H}$. Let $\mathcal{H}$ be a hyperbolic conjugacy class in $\Gamma$ and $\ell$ the $\mathcal{H}$-invariant closed geodesic on the surface. The main asymptotic for the counting function of the orb...
For $\Gamma$ a cocompact or cofinite Fuchsian group, we study the lattice
point problem on the Riemann surface $\Gamma\backslash\mathbb{H}$. The main
asymptotic for the counting of the orbit $\Gamma z$ inside a circle of radius
$r$ centered at $z$ grows like $c e^r$. Phillips and Rudnick studied
$\Omega$-results for the error term and mean results...
For $\Gamma$ a cocompact or cofinite Fuchsian group, we study the hyperbolic
lattice point problem in conjugacy classes, which is a modification of the
classical hyperbolic lattice point problem. We use large sieve inequalities for
the Riemann surfaces $\Gamma\backslash \mathbb H$ to obtain average results for
the error term, which are conjecturall...