Related papers: Large Weyl sums and Hausdorff dimension
We examine a family of three-dimensional exponential sums with monomials and provide estimates which are in some instances sharper than those stemming from approaches entailing the use of existing bounds pertaining to analogous sums.
Let $SL_2$ be the rank one simple algebraic group defined over an algebraically closed field $k$ of characteristic $p>0$. The paper presents a new method for computing the dimension of the cohomology spaces $\text{H}^n(SL_2,V(m))$ for Weyl…
In this paper, we investigate the Hausdorff dimension of the invariant measures of the iterated function system (IFS) $\{\alpha x, \beta x, \gamma x+(1-\gamma)\}$. We provide an "almost every" type result by a direct application of the…
Let $\zeta$ be a real transcendental number. We introduce a new method to find upper bounds for the classical exponent $\widehat{w}_{n}(\zeta)$ concerning uniform polynomial approximation. Our method is based on the parametric geometry of…
The main goal of our paper is to establish a connection between the Weyl modules of the current Lie superalgebras (twisted and untwisted) attached to $\mathfrak{osp}(1,2)$ and the nonsymmetric Macdonald polynomials of types $A_2^{(2)}$ and…
We establish new quantitative estimates for general systems of functions with wavelet-type dyadic structure. These estimates are applied to obtain the optimal growth of various types of Weyl multipliers for certain wavelet-type systems.…
We look at the rate of growth of the partial quotients of the infinite continued fraction expansion of an irrational number relative to the rate of approximation of the number by its convergents. In non-generic cases the Hausdorff dimension…
It is well known that the classical Gauss sum, normalized by the square-root number of terms, takes only finitely many values. If one restricts the range of summation to a subinterval, a much richer structure emerges. We prove a limit law…
We study the computational complexity of determining the Hausdorff distance of two polytopes given in halfspace- or vertex-presentation in arbitrary dimension. Subsequently, a matching problem is investigated where a convex body is allowed…
Let F be an algebraically closed field of positive characteristic p. The third author and Will Turner gave an explicit description of the extension algebra of Weyl modules for GL_2(F). This, in particular, produced an explicit basis. We…
We prove bounds for the absolute sum of all level-$k$ Fourier coefficients for $(-1)^{p(x)}$, where polynomial $p:\mathbf{F}_2^n \to \mathbf{F}_2$ is of degree $1$ or degree $2$.
We study Kloosterman sums on the orthogonal groups $SO_{3,3}$ and $SO_{4,2}$, associated to short elements of their respective Weyl groups. An explicit description for these sums is obtained in terms of multi-dimensional exponential sums.…
In this paper, we study Hausdorff and Fourier dimension from the point of view of effective descriptive set theory and Type-2 Theory of Effectivity. Working in the hyperspace $\mathbf{K}(X)$ of compact subsets of $X$, with $X=[0,1]^d$ or…
The Weyl modules in the sense of V.Chari and A.Pressley [CP] over the current Lie algebra on an affine variety are studied. We show that local Weyl modules are finite-dimensional and generalize the tensor product decomposition theorem from…
In [14], the authors developed a new approach to the computation of the Hausdorff dimension of the invariant set of an iterated function system or IFS. In this paper, we extend this approach to incorporate high order approximation methods.…
Recently, Stull [18], [17] resolved a long-standing open problem posed by Lutz, on whether the set of effective Hausdorff dimensions of points on a straight line in $\mathbb{R}^2$ -- the effective dimension spectrum of the line -- contains…
We present estimates for smooth Weyl sums of use on sets of major arcs in applications of the Hardy-Littlewood method. In particular, we derive mean value estimates on major arcs for smooth Weyl sums of degree $k$ delivering essentially…
A theorem is proved concerning approximation of analytic functions by multivariate polynomials in the $s$-dimensional hypercube. The geometric convergence rate is determined not by the usual notion of degree of a multivariate polynomial,…
Classes of polynomial differential equations of degree n are considered. An explicit upper bound on the size of the coefficients are given which implies that each equation in the class has exactly n complex periodic solutions. In most of…
We compute the Hausdorff dimension of the set of $\psi$-exactly approximable vectors, in the simultaneous case, in dimension strictly larger than $2$ and for approximating functions $\psi$ with order at infinity less than or equal to $-2$.…