Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userExtension to the Beraha-Kahane-Weiss Theorem with Applications
63
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
The beautiful Beraha-Kahane-Weiss theorem has found many applications within graph theory, allowing for the determination of the limits of root of graph polynomials in settings as vast as chromatic polynomials, network reliability, and generating polynomials related to independence and domination. Here we extend the class of functions to which the BKW theorem can be applied, and provide some applications in combinatorics.
rate research
Read More
The Friedgut-Kalai-Naor (FKN) theorem states that if $f$ is a Boolean function on the Boolean cube which is close to degree 1, then $f$ is close to a dictator, a function depending on a single coordinate. The author has extended the theorem to the slice, the subset of the Boolean cube consisting of all vectors with fixed Hamming weight. We extend the theorem further, to the multislice, a multicoloured version of the slice. As an application, we prove a stability version of the edge-isoperimetric inequality for settings of parameters in which the optimal set is a dictator.
A celebrated result of Morse and Hedlund, stated in 1938, asserts that a sequence $x$ over a finite alphabet is ultimately periodic if and only if, for some $n$, the number of different factors of length $n$ appearing in $x$ is less than $n+1$. Attempts to extend this fundamental result, for example, to higher dimensions, have been considered during the last fifteen years. Let $dge 2$. A legitimate extension to a multidimensional setting of the notion of periodicity is to consider sets of $ZZ^d$ definable by a first order formula in the Presburger arithmetic $<ZZ;<,+>$. With this latter notion and using a powerful criterion due to Muchnik, we exhibit a complete extension of the Morse--Hedlund theorem to an arbitrary dimension $d$ and characterize sets of $ZZ^d$ definable in $<ZZ;<,+>$ in terms of some functions counting recurrent blocks, that is, blocks occurring infinitely often.
Generalized polynomials are mappings obtained from the conventional polynomials by the use of operations of addition, multiplication and taking the integer part. Extending the classical theorem of H. Weyl on equidistribution of polynomials, we show that a generalized polynomial $q(n)$ has the property that the sequence $(q(n) lambda)_{n in mathbb{Z}}$ is well distributed $bmod , 1$ for all but countably many $lambda in mathbb{R}$ if and only if $limlimits_{substack{|n| rightarrow infty n otin J}} |q(n)| = infty$ for some (possibly empty) set $J$ having zero density in $mathbb{Z}$. We also prove a version of this theorem along the primes (which may be viewed as an extension of classical results of I. Vinogradov and G. Rhin). Finally, we utilize these results to obtain new examples of sets of recurrence and van der Corput sets.
We give a new proof of the Cauchy-Davenport Theorem for linear maps given by Herdade et al., (2015). This theorem gives a lower bound on the size of the image of a linear map on a grid. Our proof is purely combinatorial and offers a partial insight into the range of parameters not handled previously.
In this paper, we prove an extension theorem for spheres of square radii in $mathbb{F}_q^d$, which improves a result obtained by Iosevich and Koh (2010). Our main tool is a new point-hyperplane incidence bound which will be derived via a cone restriction theorem. We also will study applications on distance problems.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
