The Modulus of Polynomials with Zeros on the Unit Circle: A Problem of Erdos

Abstract

must tend to infinity. (Note that this is in fact Problem 21 in Erdds' very first Some unsolved problems paper; see Michigan Math. J. 4 (1957), 291-300.) Let us first say a few words on the motivation behind this problem. If the points 21+ ... I an are just the nth roots of unity, i.e., tj = e27ij/nI i = 1 <, < n, then Pn(z) = n1, andsoAn =2.Thisshowsthatthe conjecture lim sup An = ?? must be a consequence of the impossibility of getting every segment 1, ;2, . n close to uniform distribution. There seems, therefore, to be an intimate connection with the classical van der Corput problem of discrepancy of sequences in the unit interval. By changing variables, one can easily pass from the unit circle to the unit interval as follows. Consider the function