Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOn the Lower Bound of the Divisibility of Exponential Sums in Binomial Case
171
0
0.0
(
0
)
Added by
Xiaogang Liu
Publication date
2015
fields
Informatics Engineering
and research's language is
English
Authors
Xiaogang Liu
Ask ChatGPT about the research
No Arabic abstract
Francis Castro, et al [2] computed the exact divisibility of families of exponential sums associated to binomials $F(X) = aX^{d_1} + bX^{d_2}$ over $mathbb{F}_p$, and a conjecture is presented for related work. Here we study this question.
rate research
Read More
We propose higher-order generalizations of Jacobsthals $p$-adic approximation for binomial coefficients. Our results imply explicit formulae for linear combinations of binomial coefficients $binom{ip}{p}$ ($i=1,2,dots$) that are divisible by arbitrarily large powers of prime $p$.
We prove that the Hausdorff dimension of the set $mathbf{x}in [0,1)^d$, such that $$ left|sum_{n=1}^N expleft(2 pi ileft(x_1n+ldots+x_d n^dright)right) right|ge c N^{1/2} $$ holds for infinitely many natural numbers $N$, is at least $d-1/2d$ for $d ge 3$ and at least $3/2$ for $d=2$, where $c$ is a constant depending only on $d$. This improves the previous lower bound of the first and third authors for $dge 3$. We also obtain similar bounds for the Hausdorff dimension of the set of large sums with monomials $xn^d$.
Recently, Ni and Pan proved a $q$-congruence on certain sums involving central $q$-binomial coefficients, which was conjectured by Guo. In this paper, we give a generalization of this $q$-congruence and confirm another $q$-congruence, also conjectured by Guo. Our proof uses Ni and Pans technique and a simple $q$-congruence observed by Guo and Schlosser.
In this paper, we derive a useful lower bound for the Kullback-Leibler divergence (KL-divergence) based on the Hammersley-Chapman-Robbins bound (HCRB). The HCRB states that the variance of an estimator is bounded from below by the Chi-square divergence and the expectation value of the estimator. By using the relation between the KL-divergence and the Chi-square divergence, we show that the lower bound for the KL-divergence which only depends on the expectation value and the variance of a function we choose. This lower bound can also be derived from an information geometric approach. Furthermore, we show that the equality holds for the Bernoulli distributions and show that the inequality converges to the Cram{e}r-Rao bound when two distributions are very close. We also describe application examples and examples of numerical calculation.
In this note, a criterion for a class of binomials to be permutation polynomials is proposed. As a consequence, many classes of binomial permutation polynomials and monomial complete permutation polynomials are obtained. The exponents in these monomials are of Niho type.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
