Erdos Problem About Bad Denominators and Its Combinatorial Solution

Explore a 2-hour seminar lecture from the Kolmogorov Seminar series that delves into a fascinating mathematical problem concerning rational approximations and word combinatorics. Learn about the Erdős problem of finding real numbers that resist good rational approximations when restricted to specific denominator sequences. Discover how while Dirichlet's theorem guarantees increasingly precise rational approximations for any real number using unrestricted denominators, the situation changes dramatically when limited to sparse sequences of denominators. Follow the innovative approach developed by Matthieu Rosenfeld from LIRMM, who provides an elegant solution using word combinatorics techniques, particularly Joe Miller's potential argument for avoiding forbidden strings, rather than traditional number theory methods. Examine specific examples like the binary representation of 1/3 to understand how certain denominator sequences fail to provide good approximations, and explore the mathematical conditions that make this phenomenon possible.