SUBBLOCK OCCURRENCES IN SIGNED DIGIT ... - CiteSeerX
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SUBBLOCK OCCURRENCES IN SIGNED DIGIT ... - CiteSeerX
true, and 0 otherwise. With this notation we can count the number of subblock occurrences of b in (the symmetric signed digit expansions of) n via. (1.1) sb(n) =.
SUBBLOCK OCCURRENCES IN SIGNED DIGIT REPRESENTATIONS PETER J. GRABNER† , CLEMENS HEUBERGER‡ , AND HELMUT PRODINGER∗ Abstract. Signed digit representations with base q and digits − 2q , . . . , 2q (and uniqueness being enforced by applying a special rule which decides whether −q/2 or q/2 should be taken) are considered with respect to counting the occurrences of a given (contiguous) subblock of length r. The average number of occurrences amongst the numbers 0, . . . , n−1 turns out to be const · logq n + δ(logq n) + o(1), with a constant and a periodic function of period one depending on the given subblock; they are explicitly described. Furthermore, we use probabilistic techniques to prove a central limit theorem for the number of occurrences of a given subblock.
1. Introduction If we write 106 in binary we obtain 11110100001001000000. This word contains the (contiguous) substring 100 three times. In this paper we are concerned with counting occurrences of a given substring (or block) (like 100) in representations of numbers. Since this is typically somewhat erratic, we are interested in an average 1 X (number of occurrences of a given block w in the representation of n). N 0≤n
q +1 u+v 2 − m + 1 ≥ 1 − = ξq2 −1 q2 q 2 (q + 1)
by (2.6). This also implies ε0 (x) = ε0 (u) as requested. In order to deal with ℓ ≥ 1 we consider X X (εk (x) − εk (u))q k = v − εk (x)q k . k≥0
