Related papers: On the balanced Voronoi formula for GL$_N$

We consider the bilinear Schroedinger equation on a bounded one-dimensional domain and we provide explicit times such that the global exact controllability is verified. In addition, we show how to construct controls for the global…

We consider a system of an arbitrary number of \textsc{1d} linear Schr\"odinger equations on a bounded interval with bilinear control. We prove global exact controllability in large time of these $N$ equations with a single control. This…

Let $K_{q^n}(a)$ be a Kloosterman sum over the finite field $\F_{q^n}$ of characteristic $p$. In this note so called subfield conjecture is proved in case $p>3$: if $a\ne0$ belongs to the proper subfield $\F_q$ of $\F_{q^n}$, then…

We give nontrivial bounds for the bilinear sums $$ \sum_{u = 1}^{U} \sum_{v=1}^V \alpha_u \beta_v \mathbf{\,e}_p(u/f(v)) $$ where $\mathbf{\,e}_p(z)$ is a nontrivial additive character of the prime finite field ${\mathbb F}_p$ of $p$…

Problems of (i) precise (exact) bound for families of hyperelliptic curves over prime finite fields and (ii) equidistribution of angles of Kloosterman sums are discussed.

Let $(\mathcal{G},\nu)$ be a $t$-discrete ergodic groupoid. Consider a finite Von Neumann algebra $\mathcal{M}$ with separable predual. We prove that every uniformly bounded measurable representation $\rho:\mathcal{G} \rightarrow…

We introduce an algebra model to study higher order sum rules for orthogonal polynomials on the unit circle. We build the relation between the algebra model and sum rules, and prove an equivalent expression on the algebra side for the sum…

Quantitative (or controlled) $K$-theory for $C^*$-algebras was used by Guoliang Yu in his work on the Novikov conjecture, and later developed more formally by Yu together with Herv\'e Oyono-Oyono. In this paper, we extend their work by…

We prove non-trivial bounds for bilinear forms with hyper-Kloosterman sums with characters modulo a prime $q$ which, for both variables of length $M$, are non-trivial as soon as $M\geq q^{3/8+\delta}$ for any $\delta>0$. This range, which…

We report on computations of the cohomology of GL_2(O_D) and SL_2(O_D), where D<0 is a fundamental discriminant. These computations go well beyond earlier results of Vogtmann and Scheutzow. We use the technique of homology of Voronoi…

We develop an explicit version of the Kuznetsov trace formula for GSp(4), relating sums of Fourier coefficients to Kloosterman sums. We study the precise analytic behaviour of both the spectral and the arithmetic transforms arising in the…

We consider the distribution of the polygonal paths joining partial sums of classical Kloosterman sums, as their parameter varies modulo a prime tending to infinity. Using independence of Kloosterman sheaves, we prove convergence in the…

We prove, using combinatorics and Kloosterman sum technology that if $A \subset {\Bbb F}_q$, a finite field with $q$ elements, and $q^{{1/2}} \lesssim |A| \lesssim q^{{7/10}}$, then $\max \{|A+A|, |A \cdot A|\} \gtrsim…

We review the model of the Voronoi Diagrams which allows to reproduce the large-scale structures of our universe as given by the astronomical catalogs. The observed number of galaxies in a given solid angle with a chosen flux/magnitude…

We prove a new case of spectral reciprocity formulae for the product of $GL(n+1) \times GL(n)$ and $GL(n) \times GL(n-1)$ Rankin-Selberg $L$-functions ($n \geq 3$), which are first developed by Blomer and Khan in \cite{BK17} for degree 8…

Exponential sums have applications to a variety of scientific fields, including, but not limited to, cryptography, coding theory and information theory. Closed formulas for exponential sums of symmetric Boolean functions were found by Cai,…

Let F be a non-archimedean local field of characteristic zero. We consider distributions on GL(n+1,F) which are invariant under the adjoint action of GL(n,F). We prove that any such distribution is invariant with respect to transposition.…

We prove approximate controllability of the bilinear Schr\"odinger equation in the case in which the uncontrolled Hamiltonian has discrete non-resonant spectrum. The results that are obtained apply both to bounded or unbounded domains and…

Let $I$ be a segment in the $d$-dimensional Euclidean space $\mathbb E^d$. Let $P$ and $P+I$ be parallelohedra in $\mathbb E^d$, where "+" denotes the Minkowski sum. We prove that Voronoi's Conjecture holds for $P+I$, i.e. $P+I$ is a…

We consider the distribution of polygonal paths joining the partial sums of normalized Kloosterman sums modulo an increasingly high power p^n of a fixed odd prime p, a pure depth-aspect analogue of theorems of Kowalski-Sawin and…