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We consider Chebyshev polynomials, $T_n(z)$, for infinite, compact sets$\frak{e} \subset \mathbb{R}$ (that is, the monic polynomials minimizing thesup-norm, $\Vert T_n \Vert_{\frak{e}}$, on $\frak{e}$). We resolve a $45+$ yearold conjecture of Widom that for finite gap subsets of $\mathbb{R}$, hisconjectured asymptotics (which we call Szeg\H{o}-Widom asymptotics) holds. Wealso prove the first upper bounds of the form $\Vert T_n \Vert_{\frak{e}} \leqQ C({\frak{e}})^n$ (where $C(\frak{e})$ is the logarithmic capacity of$\frak{e}$) for a class of $\frak{e}$'s with an infinite number of components,explicitly for those $\frak{e} \subset \mathbb{R}$ that obey a Parreau-Widomcondition.

Asymptotics of Chebyshev Polynomials, I. Subsets of $\mathbb{R}$