Yuveshen Mooroogen is a master’s student at University of British Columbia, advised by Dr. Malabika Pramanik. Yuveshen is expected to graduate this year and intends to start his doctoral studies in September 2023.
Abstract: An arithmetic progression (AP) is a collection of equally-spaced real numbers. It may be finite or countably infinite. It is known that if a subset of the real line has positive Lebesgue measure, then it contains a k-term AP for every natural number k. In joint work with Laurestine Bradford and Hannah Kohut, we prove that this result does not extend to infinite APs in the following sense: for each real number p in [0,1), we construct a subset of the real line that intersects every interval of unit length in a set of measure at least p, but that does not contain any infinite AP. In this talk, I explain the geometric features of our set that allow it to avoid such progressions. I will also discuss two recent preprints, due to Kolountzakis–Papageorgiou and to Burgin–Goldberg–Keleti–MacMahon–Wang, that were inspired by our work. Our article is available on ArXiv: https://urldefense.com/v3/__https://arxiv.org/abs/2205.04786__;!!IBzWLUs!Sl74VoDvQ3Fi2bNuQkpvscSnyrANF36fZYvN8T73i-g_GMN6nIWkAwYyfzOLuxjUzYdHdSfQpuQP5C3WI55U_Spr3Gs$ . It was recently accepted for publication in Proceedings of the American Mathematical Society.