arXiv subjects

Chris Freiling

Publications and source records attributed to Chris Freiling.

5 recordsLinked to original sources

arXiv2026

The MZ-property for three-term differences

We study the Marcinkiewicz--Zygmund property for finite differences. A difference \[ D(f,h)=\sum_{k=1}^n a_k f(b_kh) \] is said to have the MZ-property if, for every measurable function $f$, \[ D(f,h)=o(h) \quad\text{and}\quad f(h)=o(1) \] imply \[ f(h)=o(h). \] In this paper, we characterize all three-node differences \[ D(f,h)=af(qh)+bf(rh)+cf(sh) \] having the MZ-property. In particular, for irrational differences with distinct nodes, we obtain explicit necessary and sufficient conditions in terms of the coefficients $a,b,c$ and the nodes $q,r,s$. As an application, our results resolve an open problem concerning a second-order Riemann difference.

math.CA↗

arXiv2021

On Levine's notorious hat puzzle

The Levine hat game requires $n$ players, each wearing an infinite random stack of black and white hats, to guess the location of a black hat on their own head seeing only the hats worn by all the other players. They are allowed a strategy session before the game, but no further communication. The players collectively win if and only if all their guesses are correct. In this paper we give an overview of what is known about strategies for this game, including an extended discussion of the case with $n = 2$ players (and a conjecture for an optimal strategy in this case). We also prove that $V_n$, the optimal value of the joint success probability in the $n$-player game, is a strictly decreasing function of $n$.

math.CO↗

arXiv2013

Achievable Rate Regions for Network Coding

Determining the achievable rate region for networks using routing, linear coding, or non-linear coding is thought to be a difficult task in general, and few are known. We describe the achievable rate regions for four interesting networks (completely for three and partially for the fourth). In addition to the known matrix-computation method for proving outer bounds for linear coding, we present a new method which yields actual characteristic-dependent linear rank inequalities from which the desired bounds follow immediately.

cs.IT↗

arXiv2011

Non-Shannon Information Inequalities in Four Random Variables

Any unconstrained information inequality in three or fewer random variables can be written as a linear combination of instances of Shannon's inequality I(A;B|C) >= 0 . Such inequalities are sometimes referred to as "Shannon" inequalities. In 1998, Zhang and Yeung gave the first example of a "non-Shannon" information inequality in four variables. Their technique was to add two auxiliary variables with special properties and then apply Shannon inequalities to the enlarged list. Here we will show that the Zhang-Yeung inequality can actually be derived from just one auxiliary variable. Then we use their same basic technique of adding auxiliary variables to give many other non-Shannon inequalities in four variables. Our list includes the inequalities found by Xu, Wang, and Sun, but it is by no means exhaustive. Furthermore, some of the inequalities obtained may be superseded by stronger inequalities that have yet to be found. Indeed, we show that the Zhang-Yeung inequality is one of those that is superseded. We also present several infinite families of inequalities. This list includes some, but not all of the infinite families found by Matus. Then we will give a description of what additional information these inequalities tell us about entropy space. This will include a conjecture on the maximum possible failure of Ingleton's inequality. Finally, we will present an application of non-Shannon inequalities to network coding. We will demonstrate how these inequalities are useful in finding bounds on the information that can flow through a particular network called the Vamos network.

cs.IT↗

arXiv2010

Linear rank inequalities on five or more variables

Ranks of subspaces of vector spaces satisfy all linear inequalities satisfied by entropies (including the standard Shannon inequalities) and an additional inequality due to Ingleton. It is known that the Shannon and Ingleton inequalities generate all such linear rank inequalities on up to four variables, but it has been an open question whether additional inequalities hold for the case of five or more variables. Here we give a list of 24 inequalities which, together with the Shannon and Ingleton inequalities, generate all linear rank inequalities on five variables. We also give a partial list of linear rank inequalities on six variables and general results which produce such inequalities on an arbitrary number of variables; we prove that there are essentially new inequalities at each number of variables beyond four (a result also proved recently by Kinser).

cs.IT↗