On the Characterization of Canonical Number Systems

It is well known that each positive integer n can be expressed uniquely as a sum n = d0 + d1b + . . . + dhb h with an integral base number b ≥ 2, dh 6= 0 and di ∈ {0, . . . , b − 1}. This concept can be generalized in several directions. On the one hand the base sequence 1, b, b, . . . can be replaced by a sequence 1 = u0 < u1 < u2 < . . . to obtain representations of positive integers. Of special interest is the case where the sequence {ui}i=0 is defined by a linear recurrence. A famous example belonging to this class is the so-called Zeckendorf representation. On the other hand, one can generalize the set of numbers which can be represented. We mention two kinds of number systems belonging to this class: The so called β-expansions introduced by Rényi [27] which are representations of real numbers in the unit interval as sums of powers of a real base number β. These digit representations of real numbers are strongly related to digit representations of positive integers if β is a zero of the characteristic polynomial of a linear recurring base sequence {ui}i=0. Of special interest is the case where β is a Pisot number. These expansions have been extensively studied. We mention here the papers Berend-Frougny [6] , Frougny [12, 13], Frougny-Solomyak [14, 15] and Loraud [25] and refer to the references given there. Another kind of number systems which admit the representation of a set which is different from N are the so-called canonical number systems (for short CNS). Since CNS form the main object studied in the present paper we recall their definition (cf. Akiyama-Pethő [2]).