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Seminar Qu茅bec-Vermont Number Theory

Title: Breaking the logarithmic barrier in Roth's theorem

Abstract:We present an improvement to Roth's theorem on arithmetic progressions, implying the first non-trivial case of a conjecture of Erd艖s: if a subset A of {1,2,3,...} is not too sparse, in that the sum of its reciprocals diverges, then A must contain infinitely many three-term arithmetic progressions. Although a problem in number theory and combinatorics on the surface, it turns out to have fascinating links with geometry, harmonic analysis and probability, and we shall aim to give something of a flavour of this.

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