On the weighted sum of consecutive values of an additive representation function

2019 ◽  
Vol 95 (1-2) ◽  
pp. 115-121
Author(s):  
Xiao-Hui Yan ◽  
Yong-Gao Chen
Keyword(s):  
Weighted Sum ◽  
Representation Function ◽  
Additive Representation ◽  
Additive Representation Function

On the k-th difference of an additive representation function

2011 ◽  
Vol 48 (1) ◽  
pp. 93-103
Author(s):  
Sándor Kiss
Keyword(s):  
Infinite Sequence ◽  
Probabilistic Methods ◽  
Number Of Solutions ◽  
Fixed Integer ◽  
Representation Function ◽  
Additive Representation ◽  
Consecutive Integers ◽  
Additive Representation Function ◽  
Positive Integers

Let k ≧ 2 be a fixed integer, A = {a1, a2, …} (a1 < a2 < …) be an infinite sequence of positive integers, and let Rk(n) denote the number of solutions of \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$a_{i_1 } + a_{i_2 } + \cdots + a_{i_k } = n,a_{i_1 } \in \mathcal{A},...,a_{i_k } \in \mathcal{A}$$ \end{document}. Let B(A, N) denote the number of blocks formed by consecutive integers in A up to N. In [5], it was proved that if k > 2 and \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\lim _{N \to \infty } \frac{{B(\mathcal{A},N)}}{{\sqrt[k]{N}}}$$ \end{document} = ∞ then |δl(Rk(n))| cannot be bounded for l ≦ k. The aim of this paper is to show that the above result is nearly best possible. We are using probabilistic methods.

scholarly journals On an additive representation function

2004 ◽  
Vol 104 (2) ◽  
pp. 327-334 ◽  
Author(s):  
R. Balasubramanian ◽  
Gyan Prakash
Keyword(s):  
Representation Function ◽  
Additive Representation ◽  
Additive Representation Function

Hypoelliptic Laplacian and Orbital Integrals (AM-177)

2011 ◽  
Author(s):  
Jean-Michel Bismut
Keyword(s):  
Fixed Point ◽  
Symmetric Space ◽  
Heat Kernel ◽  
Geodesic Flow ◽  
Casimir Operator ◽  
Index Theory ◽  
Weighted Sum ◽  
Orbital Integrals ◽  
Hypoelliptic Heat Kernel ◽  
Wave Kernel

This book uses the hypoelliptic Laplacian to evaluate semisimple orbital integrals in a formalism that unifies index theory and the trace formula. The hypoelliptic Laplacian is a family of operators that is supposed to interpolate between the ordinary Laplacian and the geodesic flow. It is essentially the weighted sum of a harmonic oscillator along the fiber of the tangent bundle, and of the generator of the geodesic flow. In this book, semisimple orbital integrals associated with the heat kernel of the Casimir operator are shown to be invariant under a suitable hypoelliptic deformation, which is constructed using the Dirac operator of Kostant. Their explicit evaluation is obtained by localization on geodesics in the symmetric space, in a formula closely related to the Atiyah-Bott fixed point formulas. Orbital integrals associated with the wave kernel are also computed. Estimates on the hypoelliptic heat kernel play a key role in the proofs, and are obtained by combining analytic, geometric, and probabilistic techniques. Analytic techniques emphasize the wavelike aspects of the hypoelliptic heat kernel, while geometrical considerations are needed to obtain proper control of the hypoelliptic heat kernel, especially in the localization process near the geodesics. Probabilistic techniques are especially relevant, because underlying the hypoelliptic deformation is a deformation of dynamical systems on the symmetric space, which interpolates between Brownian motion and the geodesic flow. The Malliavin calculus is used at critical stages of the proof.

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