arXiv subjects
Publications and source records attributed to Guy Lachman.
Let $S_k$ be a sequence of compact hyperbolic surfaces of increasing volume which locally converges to a random rooted surface. We show that if the normalized sum of the reciprocal lengths of very short simple closed geodesics converges to 0, then the normalized logarithm of the determinant of the Laplacian of $S_k$ converges to a constant depending only the law of the limiting surface.
Let $S$ be a compact hyperbolic surface of genus $g\geq 2$ and let $I(S) = \frac{1}{\mathrm{Vol}(S)}\int_{S} \frac{1}{\mathrm{Inj}(x)^2 \wedge 1} dx$, where $\mathrm{Inj}(x)$ is the injectivity radius at $x$. We prove that for any $k\in \{1,\ldots, 2g-3\}$, the $k$-th eigenvalue $\lambda_k$ of the Laplacian satisfies \begin{equation*} \lambda_k \geq \frac{c k^2}{I(S) g^2} \, , \end{equation*} where $c>0$ is some universal constant. We use this bound to prove the heat kernel estimate \begin{equation*} \frac{1}{\mathrm{Vol}(S)} \int_S \Big| p_t(x,x) -\frac{1}{\mathrm{Vol}(S)} \Big | ~dx \leq C \sqrt{ \frac{I(S)}{t}} \qquad \forall t \geq 1 \, , \end{equation*} where $C<\infty$ is some universal constant. These bounds are optimal in the sense that for every $g\geq 2$ there exists a compact hyperbolic surface of genus $g$ satisfying the reverse inequalities with different constants.
We introduce a novel concept in topological dynamics, referred to as $k$-divergence, which extends the notion of divergent orbits. Motivated by questions in the theory of inhomogeneous Diophantine approximations, we investigate this notion in the dynamical system given by a certain flow on the space of unimodular lattices in $\mathbb{R}^d$. Our main result is the existence of $k$-divergent lattices for any $k\geq 0$. In fact, we utilize the emerging theory of parametric geometry of numbers and calculate the Hausdorff dimension of the set of $k$-divergent lattices.
$k$-level correlation is a local statistic of sequences modulo 1, describing the local spacings of $k$-tuples of elements. For $k = 2$ this is also known as pair correlation. We show that there exists a well spaced increasing sequence of reals with additive energy of order $N^3$ and Poissonian $k$-level correlation for all integers $k \geq 2$, answering in the affirmative a question raised by Aistleitner, El-Baz, and Munsch. The construction is probabilistic, and so we do not obtain a specific sequence satisfying this condition. To prove this, we show that random perturbations of a sequence with small discrepancy gives, almost surely, a sequence with Poissonian $k$-level correlation, a fact which may be of independent interest.