Kenneth A. Ross

Abstract Harmonic Analysis

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p-Sidon sets

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Hypergroup deformations and Markov chains

Let G be a compact Abelian group with character group X. Bozejko and Pytlik [Colloq. Math. 25 (1972), 117–124] introduced and studied several special types of lacunary subsets of X. This paper is based upon a hitherto unpublished detailed study of those types that most resemble Sidon sets, which the present authors had independently introduced and studied under the name of p-Sidon sets. Some, but not all, aspects of the theory of Sidon (= 1-Sidon) sets carry over to the more general setting. In Section 1 some properties of sets equivalent to p-Sidonicity are given. Section 2 contains several useful consequences of p-Sidonicity; see Theorems 2.1 and 2.4 and Corollaries 2.6 and 2.7. In Section 3, it is shown that certain Λq sets also satisfy some of the consequences listed in Section 2. Nevertheless, Λq sets need not be p-Sidon sets; see Theorem 3.1. Examples of (43)-Sidon sets that are not Sidon sets are given in Section 5. The proof that these sets are (43)-Sidon sets requires a brief study of 4-norms in Varopoulos algebras; see Section 4. In Section 6, some special results for the circle group are deduced. Many of these results appear to be new even for p = 1.

Benford's Law, A Growth Industry

Persi Diaconis and Phil Hanlon in their interesting paper(4) give the rates of convergence of some Metropolis Markov chains on the cubeZd(2). Markov chains on finite groups that are actually random walks are easier to analyze because the machinery of harmonic analysis is available. Unfortunately, Metropolis Markov chains are, in general, not random walks on group structure. In attempting to understand Diaconis and Hanlons work, the authors were led to the idea of a hypergroup deformation of a finite groupG, i.e., a continuous family of hypergroups whose underlying space isG and whose structure is naturally related to that ofG. Such a deformation is provided forZd(2), and it is shown that the Metropolis Markov chains studied by Diaconis and Hanlon can be viewed as random walks on the deformation. A direct application of the Diaconis-Shahshahani Upper Bound Lemma, which applies to random walks on hypergroups, is used to obtain the rate of convergence of the Metropolis chains starting at any point. When the Markov chains start at 0, a result in Diaconis and Hanlon(4) is obtained with exactly the same rate of convergence. These results are extended toZd(3).

Hypergroups and Signed Hypergroups

Abstract Often data in the real world have the property that the first digit 1 appears about 30% of the time, the first digit 2 appears about 17% of the time, and so on with the first digit 9 appearing about 5% of the time. This phenomenon is known as Benfords law. This paper provides a simple explanation, suitable for nonmathematicians, of why Benfords law holds for data that have been growing (or shrinking) exponentially over time. Two theorems verify that Benfords law holds if the initial values and rates of growth of the data appear at random.

Stopping Strategies and Gambler's Ruin

Hypergroups, as I understand them, have been around since the early 1970’s when Charles Dunkl, Robert Jewett and Rene Spector independently created locally compact hypergroups with the purpose of doing standard harmonic analysis. As one would expect, there were technical differences in their definitions. The standard, in the non-Soviet world, became Jewett’s 101-page paper [J] because he worked out a good deal of the basic theory that people would want. Bloom and Heyer’s book [BH] is a useful report on some of the mathematics that has been done on the basis of Jewett’s axioms.

A comparison theorem on convergence rates of random walks on groups

Let’s play a game. Roll a die and you win

Integration on locally compact spaces

2 if the die shows 1 or 2. Otherwise you lose

Repeating Decimals: A Period Piece

1. Thus about one-third of the time you win

Helgason's number and lacunarity constants

2 and about two-thirds of the time you lose