The maximal number of runs in standard Sturmian words

We investigate some repetition problems for a very special class S of strings called the standard Sturmian words, which have very compact representations in terms of sequences of integers. Usually the size of this word is exponential with respect to the size of its integer sequence, hence we are dealing with repetition problems in compressed strings. An explicit formula is given for the number rho(w) of runs in a standard word w . We show that rho(w)/vertical bar w vertical bar <= 4/5 for each w is an element of S , and there is an infinite sequence of strictly growing words w(k) is an element of S such that lim(k ->infinity) rho(w(k) )/vertical bar w(k)vertical bar = 4/5 . Moreover, we show how to compute the number of runs in a standard Sturmian word in linear time with respect to the size of its compressed representation.