Jacob Christiansen: Widom factors: from real sets to complex continua

For any compact set E in the complex plane, the n-th Chebyshev polynomial is the unique monic polynomial of degree n that minimizes the supremum norm on E. It is a classical result in uniform approximation that the n-th root of this minimum norm converges to the logarithmic capacity of the set. However, the finer asymptotic behavior is captured by the so-called Widom factors: the sequence formed by the ratio of the Chebyshev norm to the n-th power of the capacity.

To provide context, we will first briefly review the theory for real subsets. A notable example is the class of period-n sets, where the corresponding Chebyshev polynomials naturally show up in the spectral theory of periodic Jacobi matrices.

Turning to the general case of complex continua (compact, connected sets), we will focus on two fundamental questions regarding the Widom factors. A natural starting point is to ask which continua share the asymptotic behavior of the unit disk, in the sense that their Widom factors converge to 1. Furthermore, we want to understand the extreme behavior of these factors: is it possible for them to become unbounded, and if not, is there an absolute upper bound on how large they can get within the class of continua?

While it is known that the Widom factors remain uniformly bounded for sets with sufficiently smooth boundaries, it remains a major open problem whether there exists any continuum for which they are unbounded. We will discuss the ongoing search for such a set, examine theoretical lower bounds for Chebyshev norms, and explore the specific pathological and fractal geometries that would be required to force the Widom factors to grow indefinitely.