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Session 1. Analytic Number Theory
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Hilbert cubes in arithmetic sets |
Rainer Dietmann, University of London, United Kingdom
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The talk is based on the joint work with Christian Elsholtz, and joint work with Christian Elsholtz and Igor Shparlinski
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A Hilbert cube is an iterated sumset of the form \(a_0 + \{0, a_1\}
+ \dots + \{0, a_d\}\). In this talk we discuss how large the
dimension \(d\) of a Hilbert cube in "interesting" arithmetic sets
such as squares at most \(N\), squarefull numbers at most \(N\) or
pure powers at most \(N\) can be. We also briefly address the
related problem of bounding the dimension of Hilbert cubes in
quadratic residues modulo a prime. The proofs of our results combine
combinatorial methods as well as Diophantine results, bounds for
character sums and an application of the larger sieve.
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