Related papers: The structure of Zeckendorf expansions
We consider the evolution of the packing of disks (representing the position of buds) that are introduced at the top of a surface which has the form of a growing stem. They migrate downwards, while conforming to three principles, applied…
As a variant of Zeckendorf's theorem, Chung and Graham proved that every positive integer can be uniquely decomposed into a sum of even-indexed Fibonacci numbers, whose coefficients are either $0, 1$, or $2$ so that between two coefficients…
Automatic sequences are not suitable sequences for cryptographic applications since both their subword complexity and their expansion complexity are small, and their correlation measure of order 2 is large. These sequences are highly…
The paper presents fundamental metrical theorems for a class of continued fraction-like expansions known as $\theta$-expansions. We first prove Khinchine's Weak Law of Large Numbers for the sum of digits, followed by the Diamond-Vaaler…
A folklore conjecture in number theory states that the only integers whose expansions in base $3,4$ and $5$ contain solely binary digits are $0, 1$ and $82000$. In this paper, we present the first progress on this conjecture. Furthermore,…
We study a space--time block code from a maximal order in the definite quaternion algebra $(-1,-3)_{\Q}$. Its embedding into $\C^{2\times 2}$ yields an Alamouti--Eisenstein code over $\Z[w]$ with full diversity, orthogonality, and…
Let $A$ be a set of natural numbers. A set $B$, a set of natural numbers, is said to be an additive complement of the set $A$ if all sufficiently large natural numbers can be represented in the form $x+y$, where $x\in A$ and $y\in B$. This…
Zeckendorf's theorem states that every positive integer can be uniquely decomposed as a sum of nonconsecutive Fibonacci numbers. The distribution of the number of summands converges to a Gaussian, and the individual measures on gaps between…
In this document we achieve exact and asymptotic enumeration of words, compositions over a finite group, and/or integer compositions characterized by local restrictions and, separately, subsequence pattern avoidance. We also count…
An interesting characterization of the Fibonacci numbers is that, if we write them as $F_1 = 1$, $F_2 = 2$, $F_3 = 3$, $F_4 = 5, ...$, then every positive integer can be written uniquely as a sum of non-adjacent Fibonacci numbers. This is…
The purpose of this paper is twofold. First, we present a conjecture to the effect that the ranks of the syzygy modules of a smooth projective variety become normally distributed as the positivity of the embedding line bundle grows. Then,…
Investigation of divisibility properties of natural numbers is one of the most important themes in the theory of numbers. Various tools have been developed over the centuries to discover and study the various patterns in the sequence of…
A beautiful theorem of Zeckendorf states that every positive integer can be uniquely decomposed as a sum of non-consecutive Fibonacci numbers $\{F_n\}$, where $F_1 = 1$, $F_2 = 2$ and $F_{n+1} = F_n + F_{n-1}$. For general recurrences…
The preferential attachment model is a natural and popular random graph model for a growing network that contains very well-connected ``hubs''. We study the higher-order connectivity of such a network by investigating the topological…
We introduce a novel generalization of deranged Bell numbers by defining the partial deranged Bell numbers $w_{n,r}$, which count the number of set partitions of $\left[ n\right] $ with exactly $r$ fixed blocks, while the remaining blocks…
A pattern of a sequence is a sequence of integer indices with each index describing the order of first occurrence of the respective symbol in the original sequence. In a recent paper, tight general bounds on the block entropy of patterns of…
This work explores new arithmetic and combinatorial structures arising from the interplay between Farey-type graphs, Fibonacci expansions, and operadic constructions. We introduce Fibonadic numbers, defined as an inverse limit under the…
Let $\beta>1$ be a non-integer. First we show that Lebesgue almost every number has a $\beta$-expansion of a given frequency if and only if Lebesgue almost every number has infinitely many $\beta$-expansions of the same given frequency.…
We establish sharp bounds for the Hausdorff dimension of sets of irrational numbers in $(0,1)$ whose digits in the $N$-expansion are either uniformly bounded or tend to infinity. For sets with digits bounded by an integer $M \ge N$, we…
We consider infinite iterated function systems $\{f_i\}_{i=1}^{\infty}$ on $[0,1]$ with a polynomially increasing contraction rate. We look at subsets of such systems where we only allow iterates $f_{i_1}\circ f_{i_2}\circ f_{i_3}\circ...$…