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On the uniform domination number of a finite simple group
Let be a finite simple group. By a theorem of Guralnick and Kantor,
contains a conjugacy class such that for each non-identity element , there exists with . Building on this deep
result, we introduce a new invariant , which we call the uniform
domination number of . This is the minimal size of a subset of conjugate
elements such that for each , there exists with . (This invariant is closely related to the total
domination number of the generating graph of , which explains our choice of
terminology.) By the result of Guralnick and Kantor, we have for some conjugacy class of , and the aim of this paper
is to determine close to best possible bounds on for each family
of simple groups. For example, we will prove that there are infinitely many
non-abelian simple groups with . To do this, we develop a
probabilistic approach, based on fixed point ratio estimates. We also establish
a connection to the theory of bases for permutation groups, which allows us to
apply recent results on base sizes for primitive actions of simple groups.Comment: 35 pages; to appear in Trans. Amer. Math. So
Approximate groups and their applications: work of Bourgain, Gamburd, Helfgott and Sarnak
This is a survey of several exciting recent results in which techniques
originating in the area known as additive combinatorics have been applied to
give results in other areas, such as group theory, number theory and
theoretical computer science. We begin with a discussion of the notion of an
approximate group and also that of an approximate field, describing key results
of Freiman-Ruzsa, Bourgain-Katz-Tao, Helfgott and others in which the structure
of such objects is elucidated. We then move on to the applications. In
particular we will look at the work of Bourgain and Gamburd on expansion
properties of Cayley graphs on SL_2(F_p) and at its application in the work of
Bourgain, Gamburd and Sarnak on nonlinear sieving problems.Comment: 25 pages. Survey article to accompany my forthcoming talk at the
Current Events Bulletin of the AMS, 2010. A reference added and a few small
changes mad
The inverse of the star-discrepancy problem and the generation of pseudo-random numbers
The inverse of the star-discrepancy problem asks for point sets of
size in the -dimensional unit cube whose star-discrepancy
satisfies where
is a constant independent of and . The first existence results in this
direction were shown by Heinrich, Novak, Wasilkowski, and Wo\'{z}niakowski in
2001, and a number of improvements have been shown since then. Until now only
proofs that such point sets exist are known. Since such point sets would be
useful in applications, the big open problem is to find explicit constructions
of suitable point sets .
We review the current state of the art on this problem and point out some
connections to pseudo-random number generators
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