1,939 research outputs found
Minimal weight expansions in Pisot bases
For applications to cryptography, it is important to represent numbers with a
small number of non-zero digits (Hamming weight) or with small absolute sum of
digits. The problem of finding representations with minimal weight has been
solved for integer bases, e.g. by the non-adjacent form in base~2. In this
paper, we consider numeration systems with respect to real bases which
are Pisot numbers and prove that the expansions with minimal absolute sum of
digits are recognizable by finite automata. When is the Golden Ratio,
the Tribonacci number or the smallest Pisot number, we determine expansions
with minimal number of digits and give explicitely the finite automata
recognizing all these expansions. The average weight is lower than for the
non-adjacent form
From Fibonacci Numbers to Central Limit Type Theorems
A beautiful theorem of Zeckendorf states that every integer can be written
uniquely as a sum of non-consecutive Fibonacci numbers
. Lekkerkerker proved that the average number of
summands for integers in is , with the
golden mean. This has been generalized to the following: given nonnegative
integers with and recursive sequence
with , and
, every positive
integer can be written uniquely as under natural constraints on
the 's, the mean and the variance of the numbers of summands for integers
in are of size , and the distribution of the numbers of
summands converges to a Gaussian as goes to the infinity. Previous
approaches used number theory or ergodic theory. We convert the problem to a
combinatorial one. In addition to re-deriving these results, our method
generalizes to a multitude of other problems (in the sequel paper \cite{BM} we
show how this perspective allows us to determine the distribution of gaps
between summands in decompositions). For example, it is known that every
integer can be written uniquely as a sum of the 's, such that every
two terms of the same (opposite) sign differ in index by at least 4 (3). The
presence of negative summands introduces complications and features not seen in
previous problems. We prove that the distribution of the numbers of positive
and negative summands converges to a bivariate normal with computable, negative
correlation, namely .Comment: This is a companion paper to Kologlu, Kopp, Miller and Wang's On the
number of summands in Zeckendorf decompositions. Version 2.0 (mostly
correcting missing references to previous literature
On Aperiodic Subtraction Games with Bounded Nim Sequence
Subtraction games are a class of impartial combinatorial games whose
positions correspond to nonnegative integers and whose moves correspond to
subtracting one of a fixed set of numbers from the current position. Though
they are easy to define, sub- traction games have proven difficult to analyze.
In particular, few general results about their Sprague-Grundy values are known.
In this paper, we construct an example of a subtraction game whose sequence of
Sprague-Grundy values is ternary and aperiodic, and we develop a theory that
might lead to a generalization of our construction.Comment: 45 page
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