Ellipsephic efficient congruencing for the Vinogradov system
Linfoot Number Theory Seminar
17th March 2021, 11:00 am – 12:00 pm
Virtual Seminar, https://bristol-ac-uk.zoom.us/j/98065478675?pwd=ZU1IT0d3RWU2S0V3SE4raXdLbHBoUT09
An ellipsephic set consists of natural numbers with digital restrictions in a given base. Such sets have a fractal structure and can be viewed as p-adic analogues of real Cantor sets. Using Wooley's nested efficient congruencing method, we bound the number of ellipsephic solutions to the Vinogradov system of general degree k; that is, the system of diagonal equations x_1^j + ... + x_s^j = y_1^j + ... + y_s^j for j from 1 to k.
In this talk, I will focus on the key step in the proof, which uses an additive property of our chosen ellipsephic sets to improve on certain congruence conditions at a low cost. I will also touch briefly on connections to harmonic analysis.