Related papers: Exact Exponent of Remainder Term of Gelfond's Digi…

We prove an analog of Lagrange's Theorem for continued fractions on the Heisenberg group: points with an eventually periodic continued fraction expansion are those that satisfy a particular type of quadratic form, and vice-versa.

In this note, we describe an interpretation of the (continuous) Fourier transform from the perspective of the Chinese Remainder Theorem. Some related issues, including a new derivation of Poisson summation formula, are discussed.

We give an Euler Maclaurin formula with remainder for the sum of the values of a smooth function on the integral points in a simple integral polytope. This formula is proved by elementary methods.

The aim of this paper is to present an explicit reduction algorithm for Hilbert modular groups over arbitrary totally real number fields. An implementation of the algorithm is available to download from [19]. The exposition is…

Let m be a positive integer and A an elementary abelian group of order q^r with r greater than or equal to 2 acting on a finite q'-group G. We show that if for some integer d such that 2^{d} is less than or equal to (r-1) the dth derived…

We give an elementary, self-contained and quick proof of Belyi's theorem. As a by-product of our proof we obtain an explicit bound for the degree of the defining number field of a Belyi surface.

The critical exponent of an infinite word $\bf x$ is the supremum, over all finite nonempty factors $f$, of the exponent of $f$. In this note we show that for all integers $k\geq 2,$ there is a binary infinite $k$-automatic sequence with…

We give a bracket polynomial expression for intermediate terms between discriminant and resultant for pair of binary forms. As an application of the bracket polynomial expression, we give an algebraic proof of the algebraic independence of…

Under a mild Lipschitz condition we prove a theorem on the existence and uniqueness of global solutions to delay fractional differential equations. Then, we establish a result on the exponential boundedness for these solutions.

In graph theory, the Szemer\'edi regularity lemma gives a decomposition of the indicator function for any graph $G$ into a structured component, a uniform part, and a small error. This result, in conjunction with a counting lemma that…

We determine the representation-finiteness of $A\otimes B$, where both $A$ and $B$ are simply connected algebras with at least three simple modules.

An asymptotic expansion formula of Riemann sums over lattice polytopes is given. The formula is an asymptotic form of the local Euler-Maclaurin formula due to Berline-Vergne. The proof given here for Delzant lattice polytopes is independent…

We consider sequences of Davenport type or Gelfond type and prove that sequences of Davenport exponent larger than $\frac{1}{2}$ are good sequences of weights for the ergodic theorem, and that the ergodic sums weighted by a sequence of…

We provide explicit series expansions for the exponential and logarithm functions attached to a rank r Drinfeld module that generalize well known formulas for the Carlitz exponential and logarithm. Using these results we obtain a procedure…

Within the framework of the theory of quaternion column-row determinants and using determinantal representations of the Moore-Penrose inverse previously obtained by the author, we get explicit determinantal representation formulas of…

We give an exponential upper bound on the probabilitywith which the denominator of the $n$th convergent in the regular continued fraction expansion stays away from the mean $\frac{n\pi^2}{12\log2}$. The exponential rate is best possible,…

On compact Riemannian manifolds, we prove a decomposition theorem for arbitrarily bounded energy sequence of solutions of a singular elliptic equation.

We prove a second main theorem for elliptic projective planes.

The classical theorem of Erd\H os \& Wintner furnishes a criterion for the existence of a limiting distribution for a real, additive arithmetical function. This work is devoted to providing an effective estimate for the remainder term under…

We prove an explicit formula for the dependence of the exponent in the fractal uncertainty principle of Bourgain-Dyatlov on the dimension and on the regularity constant for the regular set. In particular, this implies an explicit essential…