On the Structure of Sets of Large Doubling

We investigate the structure of finite sets $A \subseteq \Z$ where $|A+A|$ is large. We present a combinatorial construction that serves as a counterexample to natural conjectures in the pursuit of an "anti-Freiman" theory in additive combinatorics. In particular, we answer a question along these lines posed by O'Bryant. Our construction also answers several questions about the nature of finite unions of $B_2[g]$ and $B^\circ_2[g]$ sets, and enables us to construct a $\Lambda(4)$ set which does not contain large $B_2[g]$ or $B^\circ_2[g]$ sets.Comment: 23 pages, changed title, revised version reflects work of Meyer that we were previously unaware o

Generalized Sums over Histories for Quantum Gravity II. Simplicial Conifolds

This paper examines the issues involved with concretely implementing a sum over conifolds in the formulation of Euclidean sums over histories for gravity. The first step in precisely formulating any sum over topological spaces is that one must have an algorithmically implementable method of generating a list of all spaces in the set to be summed over. This requirement causes well known problems in the formulation of sums over manifolds in four or more dimensions; there is no algorithmic method of determining whether or not a topological space is an n-manifold in five or more dimensions and the issue of whether or not such an algorithm exists is open in four. However, as this paper shows, conifolds are algorithmically decidable in four dimensions. Thus the set of 4-conifolds provides a starting point for a concrete implementation of Euclidean sums over histories in four dimensions. Explicit algorithms for summing over various sets of 4-conifolds are presented in the context of Regge calculus. Postscript figures available via anonymous ftp at black-hole.physics.ubc.ca (137.82.43.40) in file gen2.ps.Comment: 82pp., plain TeX, To appear in Nucl. Phys. B,FF-92-

Bounds for approximate discrete tomography solutions

In earlier papers we have developed an algebraic theory of discrete tomography. In those papers the structure of the functions $f: A \to \{0,1\}$ and $f: A \to \mathbb{Z}$ having given line sums in certain directions have been analyzed. Here $A$ was a block in $\mathbb{Z}^n$ with sides parallel to the axes. In the present paper we assume that there is noise in the measurements and (only) that $A$ is an arbitrary or convex finite set in $\mathbb{Z}^n$. We derive generalizations of earlier results. Furthermore we apply a method of Beck and Fiala to obtain results of he following type: if the line sums in $k$ directions of a function $h: A \to [0,1]$ are known, then there exists a function $f: A \to \{0,1\}$ such that its line sums differ by at most $k$ from the corresponding line sums of $h$.Comment: 16 page

Measurement of inequality with a finite number of pay states : the majorization set and its applications

I am grateful to Vassily Gorbanov, Tarik Yalcin and Fabrizio Germano for extended discussions and suggestions, and to an associate editor and a reviewer for constructive comments. I also wish to thank Francesco Andreoli, Geoffrey Burton, Joe Swierzbinski, Alain Trannoy, Claudio Zoli and seminar participants at the Aix-Marseille School of Economics for discussions. I am responsible for any errors.Peer reviewedPostprin