this paper is organized as follows. In Section 2, we provide definitions and introduce notation. In Section 3, we prove a useful general zero-one law, to show that singularity and absolute continuity of the measures are the only possibilities. In Section 4, Theorem 1.1(i) is proved, while Theorem 1.1(ii) is established in Section 5. We prove a more general version of Theorem 1.2 in Section 6. In Section 7, we prove a criterion for absolute continuity, which is used to prove Theorem 1.4 in Section 8 and Theorem 1.3 in Section 9. A connection to long-range percolation and some unsolved problems are described in Section 10. 2. Definitions. Let # = {0, 1} # be the space of binary sequences. Denote by # n the n th coordinate projection from #. Endow # with the #-field H generated by {# n } n#0 and let P be a renewal measure on (#, H), that is, a measure obeying P[# 0 = 1, # n(1) = 1, . . . , # n(m) = 1] = m # i=1 u n(i)-n(i-1) , (2.1) where u n def = P[# n = 1]. We let {T k } # k=1 denote the inter-arrival times of the renewal process: If S n = inf{m > S n-1 : #m = 1} is the time of the n th renewal, then T n = S n - S n-1 . The condition (2.1) implies that T 1 , T 2 , . . . is an i.i.d. sequence. We will use f n to denote P[T 1 = n]. In the introduction we defined u n as the probability for a Markov chain # to return to its initial state at time n. If # n = 1 {#n=o} , then the Markov property guarantees that (2.1) is satisfied. Conversely, any renewal process # can be realized as the indicator of return times of a Markov chain to its initial state. (Take, for example, the chain whose value at epoch n is the time until the next renewal, and Random Coin Tossing 7 consider returns to 0.) Thus we can move freely between these points of view. For...