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:
► Arithmetic patches, weak tangents, and dimension (with H. Yu), arXiv
Bulletin of the London Mathematical Society, 50, (2018), 85-95.
I also wrote a survey article:
► Almost arithmetic progressions in the primes and other large sets, arXiv
The American Mathematical Monthly, 126, (2019), 553-558.
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:
►Approximate arithmetic structure in large sets of integers (with H. Yu), arXiv
Real Analysis Exchange (to appear).
►Improved bounds on the dimensions of sets that avoid approximate arithmetic progressions (with P. Shmerkin & A. Yavicoli), arXiv
Journal of Fourier Analysis and Applications (to appear).
► Dimensions of sets which uniformly avoid arithmetic progressions (with K. Saito & H. Yu), arXiv
International Mathematics Research Notices (IMRN), (2019), 4419-4430.
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.
Another related 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:
► A nonlinear projection theorem for Assouad dimension and applications, arXiv
Journal of the London Mathematical Society (to appear).
► Dimension growth for iterated sumsets (with D. C. Howroyd & H. Yu), arXiv
Mathematische Zeitschrift, 293, (2019), 1015-1042.
► The Fourier dimension spectrum and sumset type problems, arXiv
submitted.