ON THE MAXIMAL MODULUS OF POLYNOMIALS ON CANTOR SETS

Let 0 < p < 1 8 and consider the Cantor set C*(p) (where C*( 1 3) would be the classical Cantor set). For any sequence ω = (ξ1, ξ2, ...), ξv ε{lunate} C*(p), let Bn(ω) = maxz ε{lunate} CΠv = 1n |z - ξv|. It is shown that there exists a constant θ = θ(p), independent of ω, such that Bn(ω) > (log n)θ for almost all n (i.e., all except a sequence of density zero). An analogous theorem for the unit circle C = {|z| = 1} instead of C*(p) (with "infinitely many" instead of "almost all") was proved before by the author (Bull. London Math. Soc. 12, 1980, 81-88), solving a problem of Erdös. © 1991.