Calculating the numbers of representations and the Garsia entropy in linear numeration systems

A Fast Algorithm for Computing Binomial Coefficients Modulo Powers of Two

“…Reference [ 13 ] uses sums of binomial coefficients modulo 2 when obtaining results related to the Garsia entropy. Binomial coefficients also occur in other equations regarding information theory formulas (e.g., [ 14 , 15 ]).…”

The entropy of Cantor-like measures

“…Subsequently, the Garsia entropy was computed for various Bernoulli convolutions, first for ̺ = (1 + √ 5)/2, the golden ratio (a simple Pisot number) in [2], then for all simple Pisot numbers in [3,12], and, finally, for all algebraic integers in [1,4]. Edson, in [6], generalized these results in a different direction, considering the contraction factor 1/̺ where ̺ is the root of x 2 − ax − b with a ≥ b and a equally spaced linear contractions. This paper focuses on a different generalization, to the case where ̺ = d is an integer greater than or equal to 2 and m equally spaced contractions S j of the form…”

Confluent Parry numbers, their spectra, and integers in positive- and negative-base number systems

“…Among exceptional properties of such bases is the existence of optimal representations [7], or the fact that the infinite word u β coding the set Z β of β-integers is reversal closed and the corresponding Rauzy fractal is centrally symmetric [5]. Linear numeration systems for confluent Parry numbers are also mentioned in [9], in connection to calculating the Garsia entropy. From our point of view, the most important aspect is that any integer linear combination of non-negative powers of the base with coefficients in {0, 1, .…”