On Random Interpolation

In a recent paper Salem and Zygmund [1] proved the following result : Put 2TCv a " = avn~ _ 2n (v = 0, 1,. . ., 2n)-{-1 and denote the 99,'(t) the v-th Rademacher function. Denote by L.(t, 0) the unique trigonometric polynomial (in 0) of degree not exceeding n for Denote Mjt) = max I L,, (t, 0)1. Then for almost all t 050<2a M,, t) lim BLOCKIN~ <_ 2. (log n)a n=~ P. ERDÖS I am going to prove the following sharper THEOREM 1. For almost all t lim M"'(t)-lim M " (t)-2-~ log log n n ="~ log log 'n n Instead of Theorem 1 we shall prove the following stronger (throughout this paper cl, C 2 ,. . will denote suitable positive constants) THEOREM 2. To every cl there exists a constant C 2 = C 2 (C i) so that for n > N(C1 , C 2) the measure of the set in t for which 2 2-log log n-C 2 < M n (t) <-log log n + C 2 7r 7r is not satisfied, is less than I/n°l. Theorem 1 follows immediately from Theorem 2 by the Borel-Cantelli Lemma. Thus we only have to prove Theorem 2. First we need two simple combinatorial lemmas. Let m be a sufficiently large integer, we define for 1 < i < m (for the purpose of these lemmas) 9'.+i (t) = 9'i (t), 99-i(t) = T .-i(t) 129