Arithmetic combinatorics

An important problem in arithmetic combinatorics (and fractal geometry) is to consider how the size of a sumset compares to the size of the sets being summed. An example problem being to determine conditions under which the Hausdorff dimension of the sumset X+X strictly exceeds the Hausdorff dimension of X. One can also conisder mixed sumsets X+Y, iterated sumsets X+X+...+X = kX, or constructions where the operation of addition is replaced by something else, e.g. multiplication. I have various results in this area, for example: I am increasingly interested in discrete analogues of these questions (and other questions from fractal geometry) cast in vector spaces over finite fields. Some more information on this can be found here: Arithmetic progressions are fundamental objects in mathematics. For example, 3, 7, 11, 15, 19 is an arithmetic progression of length 5 with gap size 4. Often one is interested in finding arithmetic progressions inside a given set, or proving that certain arithmetic progressions are not present. Recall, the celebrated Green-Tao theorem which states that the prime numbers contain arbitrarily long arithmetic progressions: structure within chaos! A famous conjecture of Erdős asserts that a set of positive integers whose reciprocals form a divergent series should contain arbitrarily long arithmetic progressions. (The Green-Tao theorem is a special case of this conjecture.) It turns out that the following weak asymptotic version of this conjecture is much easier to prove and is related to dimension: a set of positive integers whose reciprocals form a divergent series gets arbitrarily close to arbitrarily long arithmetic progressions (in a natural sense). I proved this result, together with Han Yu, here: I also wrote a survey article: These results follow from the observation that getting arbitrarily close to arbitrarily long arithmetic progressions is equivalent to having Assouad dimension equal to 1 (maximal for subsets of the line). Therefore a natural question to ask is, how large can the dimension of a set be if the set uniformly avoids arithmetic progressions. Papers on this topic include: I gave an undergraduate lecture on our weak solution to the Erdős problem, called 'An analyst's take on the integers'. The video is here.
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