Monotone Subsequences in the Sequence of Fractional Parts of Multiples of an Irrational

Hammersley [7] showed that if X1, X2, . . . is a sequence of independent identically distributed random variables whose common distribution is continuous, and if ln+(ln-) denotes the length of the longest increasing (decreasing) subsequence of X1, X2, . . ., Xn, then there is a constant c such that ln-⁄n½→ c and ln+⁄n½→ c in probability, as n → ∞. Kesten [8] showed that in fact there is almost sure convergence. Logan and Shepp [11] proved that c ≧ 2, and recently Versik and Kerov [13] have announced that c = 2.