W. T. Gowers
Published 2015-09-11Version 1
This is a short survey article written for the Erd\H{o}s centennial conference in Budapest in 2013. The main two topics covered are Szemer\'edi's theorem and its ramifications, and the Erd\H{o}s discrepancy problem. There is an emphasis on what we do not yet know, so much of the article is somewhat speculative.
Comments: 21 pages, in Erdos and arithmetic progressions, in Erdos Centennial, Bolyai Society Mathematical Studies, 25, L. Lovasz, I. Z. Ruzsa, V. T. Sos eds., Springer 2013, pp. 265-287
The lattice of arithmetic progressions
A new proof of Roth's theorem on arithmetic progressions
A Distributive Lattice Connected with Arithmetic Progressions of Length Three
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