Nadav Yesha

A random ensemble of cusp forms for the full modular group is introduced. For a weight-$k$ cusp form, restricted to a compact subdomain of the modular surface, the true order of magnitude of its expected supremum is determined to be $\asymp \sqrt{\log{k}}$, in line with the conjectured bounds. Additionally, the exponential concentration of the supremum around its median is established. Contrary to the compact case, it is shown that the global expected supremum, which is attained around the cusp, grows like $k^{1/4}$, up to a logarithmic factor.

On the circle of radius R centred at the origin, consider a “thin” sector about the fixed line $$y = \alpha x$$y = α x with edges given by the lines $$y = (\alpha \pm \epsilon ) x$$y = ( α ± ϵ ) x , where $$\epsilon = \epsilon _R \rightarrow 0$$ϵ = ϵ R → 0 as $$ R \rightarrow \infty $$R → ∞ . We establish an asymptotic count for $$S_{\alpha }(\epsilon ,R)$$S α ( ϵ , R ) , the number of integer lattice points lying in such a sector. Our results depend both on the decay rate of $$\epsilon $$ϵ and on the rationality/irrationality type of $$\alpha $$α . In particular, we demonstrate that if $$\alpha $$α is Diophantine, then $$S_{\alpha }(\epsilon ,R)$$S α ( ϵ , R ) is asymptotic to the area of the sector, so long as $$\epsilon R^{t} \rightarrow \infty $$ϵ R t → ∞ for some $$ t<2 $$t < 2 .

For a real-valued sequence $(x_n)_{n=1}^\infty$, denote by $S_N(\ell)$ the number of its first $N$ fractional parts lying in a random interval of size $\ell:=L/N$, where $L=o(N)$ as $N\to\infty$. We study the variance of $S_N(\ell)$ (the number variance) for sequences of the form $x_n=αa_n$, where $(a_n)_{n=1}^\infty$ is a sequence of distinct integers. We show that if the additive energy of the sequence $(a_n)_{n=1}^\infty$ is bounded from above by $N^{5/2-\varepsilon}/L$ for some $\varepsilon>0$, then for almost all $α$, the number variance is asymptotic to $L$ (Poissonian number variance). This holds in particular for the sequence $x_n=αn^d, d\ge 2$ whenever $L=N^β$ with $0\leβ<1/2$.

We study intermediate-scale statistics for the fractional parts of the sequence $\left(\alpha a_{n}\right)_{n=1}^{\infty}$ , where $\left(a_{n}\right)_{n=1}^{\infty}$ is a positive, real-valued lacunary sequence, and $\alpha\in\mathbb{R}$ . In particular, we consider the number of elements $S_{N}\!\left(L,\alpha\right)$ in a random interval of length $L/N$ , where $L=O\!\left(N^{1-\epsilon}\right)$ , and show that its variance (the number variance) is asymptotic to L with high probability w.r.t. $\alpha$ , which is in agreement with the statistics of uniform i.i.d. random points in the unit interval. In addition, we show that the same asymptotic holds almost surely in $\alpha\in\mathbb{R}$ when $L=O\!\left(N^{1/2-\epsilon}\right)$ . For slowly growing L, we further prove a central limit theorem for $S_{N}\!\left(L,\alpha\right)$ which holds for almost all $\alpha\in\mathbb{R}$ .

Let $$\Omega {\subset } {\mathbb {R}}^2$$Ω ⊂ R 2 be a bounded planar domain, with piecewise smooth boundary $$\partial \Omega $$∂ Ω . For $$\sigma >0$$σ > 0 , we consider the Robin boundary value problem $$\begin{aligned} -\Delta f =\lambda f, \qquad \frac{\partial f}{\partial n} + \sigma f = 0 \text{ on } \partial \Omega \end{aligned}$$- Δ f = λ f , ∂ f ∂ n + σ f = 0 on ∂ Ω where $$ \frac{\partial f}{\partial n} $$∂ f ∂ n is the derivative in the direction of the outward pointing normal to $$\partial \Omega $$∂ Ω . Let $$0<\lambda ^\sigma _0\le \lambda ^\sigma _1\le \ldots $$0 < λ 0 σ ≤ λ 1 σ ≤ … be the corresponding eigenvalues. The purpose of this paper is to study the Robin–Neumann gaps $$\begin{aligned} d_n(\sigma ):=\lambda _n^\sigma -\lambda _n^0 . \end{aligned}$$d n ( σ ) : = λ n σ - λ n 0 . For a wide class of planar domains we show that there is a limiting mean value, equal to $$2{\text {length}}(\partial \Omega )/{\text {area}}(\Omega )\cdot \sigma $$2 length ( ∂ Ω ) / area ( Ω ) · σ and in the smooth case, give an upper bound of $$d_n(\sigma )\le C(\Omega ) n^{1/3}\sigma $$d n ( σ ) ≤ C ( Ω ) n 1 / 3 σ and a uniform lower bound. For ergodic billiards we show that along a density-one subsequence, the gaps converge to the mean value. We obtain further properties for rectangles, where we have a uniform upper bound, and for disks, where we improve the general upper bound.

Let $\Omega\subset \mathbb R^2$ be a bounded planar domain, with piecewise smooth boundary $\partial \Omega$. For $\sigma>0$, we consider the Robin boundary value problem \[ -\Delta f =\lambda f, \qquad \frac{\partial f}{\partial n} + \sigma f = 0 \mbox{ on } \partial \Omega \] where $ \frac{\partial f}{\partial n} $ is the derivative in the direction of the outward pointing normal to $\partial \Omega$. Let $0<\lambda^\sigma_0\leq \lambda^\sigma_1\leq \dots $ be the corresponding eigenvalues. The purpose of this paper is to study the Robin-Neumann gaps \[ d_n(\sigma):=\lambda_n^\sigma-\lambda_n^0 . \] For a wide class of planar domains we show that there is a limiting mean value, equal to $2{\rm length}(\partial\Omega)/{\rm area}(\Omega)\cdot \sigma$ and in the smooth case, give an upper bound of $d_n(\sigma)\leq C(\Omega ) n^{1/3}\sigma $ and a uniform lower bound. For ergodic billiards we show that along a density-one subsequence, the gaps converge to the mean value. We obtain further properties for rectangles, where we have a uniform upper bound, and for disks, where we improve the general upper bound.

Koksma's equidistribution theorem from 1935 states that for Lebesgue almost every $\alpha>1$, the fractional parts of the geometric progression $(\alpha^{n})_{n\geq1}$ are equidistributed modulo one. In the present paper we sharpen this result by showing that for almost every $\alpha>1$, the correlations of all finite orders and hence the normalized gaps of $(\alpha^{n})_{n\geq1}$ mod 1 have a Poissonian limit distribution, thereby resolving a conjecture of the two first named authors. While an earlier approach used probabilistic methods in the form of martingale approximation, our reasoning in the present paper is of an analytic nature and based upon the estimation of oscillatory integrals. This method is robust enough to allow us to extend our results to a natural class of sub-lacunary sequences.

A well known result in the theory of uniform distribution modulo one (which goes back to Fej\'er and Csillag) states that the fractional parts $\{n^\alpha\}$ of the sequence $(n^\alpha)_{n\ge1}$ are uniformly distributed in the unit interval whenever $\alpha>0$ is not an integer. For sharpening this knowledge to local statistics, the $k$-level correlation functions of the sequence $(\{n^\alpha\})_{n\geq1}$ are of fundamental importance. We prove that for each $k\ge2,$ the $k$-level correlation function $R_k$ is Poissonian for almost every $\alpha>4k^2-4k-1$.