Quantitative continuity estimates
23 May 2017 | 1:55 am

Suppose is a continuous (but nonlinear) map from one normed vector space to another . The continuity means, roughly speaking, that if are such that is small, then is also small (though the precise notion of “smallness” may depend on or , particularly if is not known to be uniformly continuous). If is known to […]

Generalisations of the limit functional
11 May 2017 | 8:44 pm

Suppose one has a bounded sequence of real numbers. What kinds of limits can one form from this sequence? Of course, we have the usual notion of limit , which in this post I will refer to as the classical limit to distinguish from the other limits discussed in this post. The classical limit, if […]

New bounds for Szemerédi’s theorem, III: A polylogarithmic bound for r_4(N)
5 May 2017 | 6:48 pm

Ben Green and I have (finally!) uploaded to the arXiv our paper “New bounds for Szemerédi’s theorem, III: A polylogarithmic bound for “, submitted to Mathematika. This is the sequel to two previous papers (and an erratum to the former paper), concerning quantitative versions of Szemerédi’s theorem in the case of length four progressions. This […]


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