Christopher Lutsko
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View article: Full Poissonian local statistics of slowly growing sequences
Full Poissonian local statistics of slowly growing sequences Open
Fix $\alpha >0$ . Then by Fejér's theorem $(\alpha (\log n)^{A}\,\mathrm {mod}\,1)_{n\geq 1}$ is uniformly distributed if and only if $A>1$ . We sharpen this by showing that all correlation functions, and hence the gap distribution, are Po…
View article: Diffusion of the random Lorentz process in a magnetic field
Diffusion of the random Lorentz process in a magnetic field Open
Consider the motion of a charged, point particle moving in the complement of a Poisson distribution of hard sphere scatterers in two dimensions under the effect of a fixed magnetic field. Building on, and extending a coupling method establ…
View article: Sign changes along geodesics of modular forms
Sign changes along geodesics of modular forms Open
Given a compact segment, $β$, of a cuspidal geodesic on the modular surface, we study the number of sign changes of cusp forms and Eisenstein series along $β$. We prove unconditionally a sharp lower bound for Eisenstein series along a full…
View article: Pair correlation of the fractional parts of $\alpha n^{\theta}$
Pair correlation of the fractional parts of $\alpha n^{\theta}$ Open
Fix \alpha,\theta >0 and consider the sequence (\alpha n^{\theta}\; \mathrm{mod}\; 1)_{n\ge 1} . Since the seminal work of Rudnick–Sarnak (1998) and due to the Berry–Tabor conjecture in quantum chaos, the fine-scale properties of these dil…
View article: Effective Counting in Sphere Packings
Effective Counting in Sphere Packings Open
Given a Zariski-dense, discrete group, Γ, of isometries acting on (n + 1)- dimensional hyperbolic space, we use spectral methods to obtain a sharp asymptotic formula for the growth rate of certain Γ-orbits. In particular, this allows us to…
View article: Average variance bounds for integer points on the sphere
Average variance bounds for integer points on the sphere Open
Consider the integer points lying on the sphere of fixed radius projected onto the unit sphere. Duke showed that, on congruence conditions for the radius squared, these points equidistribute. To further this study of equidistribution, we c…
View article: Polyhedral bounds on the joint spectrum and temperedness of locally symmetric spaces
Polyhedral bounds on the joint spectrum and temperedness of locally symmetric spaces Open
Given a real semisimple connected Lie group $G$ and a discrete subgroup $Γ< G$ we prove a precise connection between growth rates of the group $Γ$, polyhedral bounds on the joint spectrum of the ring of invariant differential operators, an…
View article: Counting in Lattice Orbits
Counting in Lattice Orbits Open
Given a discrete lattice, $Γ< \text{SL}_m(\mathbb{R})$, and a base point $o\in \mathbb{R}^m$, let $N_Γ(T)$ denote the number of points in the orbit $o\cdot Γ$ whose (Euclidean) length is bounded by a growing parameter, $T$. We demonstrate …
View article: Hyperbolic lattice point counting in unbounded rank
Hyperbolic lattice point counting in unbounded rank Open
We use spectral analysis to give an asymptotic formula for the number of matrices in SL(n, Z) of height at most T with strong error terms, far beyond the previous known, both for small and large rank.
View article: Norm bounds on Eisenstein series
Norm bounds on Eisenstein series Open
We study the sup-norm and mean-square-norm problems for Eisenstein series on certain arithmetic hyperbolic orbifolds, producing sharp exponents for the modular surface and Picard 3-fold. The methods involve bounds for Epstein zeta function…
View article: An abstract spectral approach to horospherical equidistribution
An abstract spectral approach to horospherical equidistribution Open
This paper introduces an abstract spectral approach to prove effective equidistribution of expanding horospheres in hyperbolic manifolds. The method, which is motivated by the approach to counting developed by (Lax-Phillips 1982), produces…
View article: Sarnak's spectral gap question
Sarnak's spectral gap question Open
We answer in the affirmative a question of Sarnak's from 2007, confirming that the Patterson-Sullivan base eigenfunction is the unique square-integrable eigenfunction of the hyperbolic Laplacian invariant under the group of symmetries of t…
View article: Full Poissonian Local Statistics of Slowly Growing Sequences
Full Poissonian Local Statistics of Slowly Growing Sequences Open
Fix $α>0$, then by Fejér's theorem $ (α(\log n)^{A}\,\mathrm{mod}\,1)_{n\geq1}$ is uniformly distributed if and only if $A>1$. We sharpen this by showing that all correlation functions, and hence the gap distribution, are Poissonian provid…
View article: Effective counting in sphere packings
Effective counting in sphere packings Open
Given a Zariski-dense, discrete group, $Γ$, of isometries acting on $(n + 1)$-dimensional hyperbolic space, we use spectral methods to obtain a sharp asymptotic formula for the growth rate of certain $Γ$-orbits. In particular, this allows …
View article: Correlations of the Fractional Parts of $αn^θ$
Correlations of the Fractional Parts of $αn^θ$ Open
Let $m\geq 3$, we prove that $(αn^θ\mod 1)_{n>0}$ has Poissonian $m$-point correlation for all $α>0$, provided $θ
View article: Pair Correlation of the Fractional Parts of $αn^θ$
Pair Correlation of the Fractional Parts of $αn^θ$ Open
Fix $α,θ>0$, and consider the sequence $(αn^θ \mod 1)_{n\ge 1}$. Since the seminal work of Rudnick--Sarnak (1998), and due to the Berry--Tabor conjecture in quantum chaos, the fine-scale properties of these dilated mononomial sequences hav…
View article: Farey Sequences for Thin Groups
Farey Sequences for Thin Groups Open
The Farey sequence is the set of rational numbers with bounded denominator. We introduce the concept of a generalized Farey sequence. While these sequences arise naturally in the study of discrete and thin subgroups, they can be used to st…
View article: Gap Statistics of the Sequence $\{α\sqrt{n}\}$
Gap Statistics of the Sequence $\{α\sqrt{n}\}$ Open
The gaps in the sequence $\{\sqrt{n}\}$ were shown by Elkies-McMullen (2004) to have a limiting distribution which is not the exponential distribution. However it is conjectured that the distribution of gaps in the sequence $\{α\sqrt{n}\}$…
View article: Gap Statistics of the Sequence $\{\alpha\sqrt{n}\}$
Gap Statistics of the Sequence $\{\alpha\sqrt{n}\}$ Open
The gaps in the sequence $\\{\\sqrt{n}\\}$ were shown by Elkies-McMullen (2004)\nto have a limiting distribution which is not the exponential distribution.\nHowever it is conjectured that the distribution of gaps in the sequence\n$\\{\\alp…
View article: Parabolic Anderson Model on R^2
Parabolic Anderson Model on R^2 Open
For my thesis project we have been studying the analysis of the parabolic Anderson model\nin 2 spatial dimensions on the whole plane, performed by Hairer and Labbe in early 2015.\nThis problem is a nice example as it requires renormalizati…