Unbounded Widom factors for orthogonal and residual polynomials

Abstract

We study Widom factors for (a) monic orthogonal polynomials in $L^2$ with respect to the equilibrium measure of a compact set $K\subset\mathbb{R}$ and (b) residual polynomials normalized at an exterior point. Using weakly equilibrium Cantor sets $K(\gamma)$, we prove: (1) Given any sequence $(c_n)$ with subexponential growth, there exists a non-polar Cantor set $K(\gamma)$, depending on $(c_n)$, such that the $L^2$ Widom factors of the associated orthogonal polynomials (with respect to the equilibrium measure of $K(\gamma)$) exceed $c_n$ for every $n$. (2) For the same $K(\gamma)$ built from $(c_n)$ and each exterior point $x_0\in\mathbb{R}\setminus K(\gamma)$, the residual Widom factors satisfy power-type lower bounds with a Harnack-distance exponent $\tau_{x_0}\in(0,1)$: they are bounded below by $c_n^{\tau_{x_0}}$ for all degrees when $x_0$ lies in an unbounded gap, and along a subsequence of degrees when $x_0$ lies in a bounded gap. Consequently, if, in addition, $(c_n)$ is monotone increasing and unbounded, then the sequence of residual Widom factors is unbounded for every $x_0\in\mathbb{R}\setminus K(\gamma)$. The proofs combine inverse-image constructions, capacity comparisons for period-$n$ sets, harmonic-measure representations for differences of Green functions, and alternation principles on nested approximants.