Asymptotics of Chebyshev polynomials, I: subsets of R

We consider Chebyshev polynomials, (Formula presented.), for infinite, compact sets (Formula presented.) (that is, the monic polynomials minimizing the (Formula presented.)-norm, (Formula presented.), on (Formula presented.)). We resolve a (Formula presented.) year old conjecture of Widom that for finite gap subsets of (Formula presented.), his conjectured asymptotics (which we call Szegő–Widom asymptotics) holds. We also prove the first upper bounds of the form (Formula presented.) (where (Formula presented.) is the logarithmic capacity of (Formula presented.)) for a class of (Formula presented.)’s with an infinite number of components, explicitly for those (Formula presented.) that obey a Parreau–Widom condition.