Linear Recurrence Sequences

Gelfond proved that for coprime 𝑏 − 1 and 𝑑 sums of digits of 𝑏-ary expressions of natural numbers are uniformly distributed on arithmetic progressions with the common difference 𝑑. Later, similar result was proved for the representations of natural numbers based on linear recurrent sequences. We consider the remainder term of the corresponding asymptotic formula and study the dichotomy between the logarithmic and power estimates of the remainder term. In the case 𝑑 = 2, we obtain some sufficient condition for the validity of the logarithmic estimate. Using them, we show that the logarithmic estimate holds for expansions based on all second-order linear recurrenct sequences and on infinite family of third-order sequences. Also we construct an example of the linear reccurent sequence of an arbitrary order with such property. On the other hand, we give an example of a third-order linear recurrent sequence for which the logarithmic estimate does not hold. We also show that for 𝑑 = 3 the logarithmic estimate does not hold even in the simplest case of the expansions based on Fibonacci numbers. In addition, we consider the representations of natural numbers based on the denominators of partial convergents of the continued fraction expansions of irrational numbers. In this case, we prove the uniformity of the distribution of sums of digits over arithmetic progressions with the common difference 2 with the logarithmic remainder term.

Let $(x_n)_{n\geq 0}$ be a linear recurrence sequence of order $k\geq 2$ satisfying $$ \begin{align*}x_n=a_1x_{n-1}+a_2x_{n-2}+\dots+a_kx_{n-k}\end{align*} $$ for all integers $n\geq k$ , where $a_1,\dots ,a_k,x_0,\dots , x_{k-1}\in \mathbb {Z},$ with $a_k\neq 0$ . In 2017, Sanna posed an open question to classify primes p for which the quotient set of $(x_n)_{n\geq 0}$ is dense in $\mathbb {Q}_p$ . In a recent paper, we showed that if the characteristic polynomial of the recurrence sequence has a root $\pm \alpha $ , where $\alpha $ is a Pisot number and if p is a prime such that the characteristic polynomial of the recurrence sequence is irreducible in $\mathbb {Q}_p$ , then the quotient set of $(x_n)_{n\geq 0}$ is dense in $\mathbb {Q}_p$ . In this article, we answer the problem for certain linear recurrence sequences whose characteristic polynomials are reducible over $\mathbb {Q}$ .

In this automata column, Jöel Ouaknine and James Worrell beautifully write about beautiful recent work on linear recurrence sequences, which are sequences of integers defined by recursion like in the Fibonacci sequence. Clearly such sequences can appear almost everywhere, so there is no particularly good reason to talk about them in the automata column, but the authors politely point out such sequences can be used to count the number of words of given length accepted by an automaton. Thanks to this connection we get to have an automata column that cites Terrence Tao and uses logarithms of algebraic numbers.

Information systems are developing and improving very fast. The requirements for synchronization for such systems also increase. In such systems linear recurrent sequences are widely used for synchronization. In this paper filtration algorithms of linear recurrent sequences sequences are synthesized. The synthesized algorithms can be classified as quasi-noncoherent (subincoherent) algorithms, as opposed to quasi-coherent algorithms, which use phase estimate of the received signal, or to noncoherent algorithms that use averaging over the entire region of phase uncertainty. The results of computer simulation of developed algorithms for various signal-to-noise ratios and accumulation time are presented.

We investigate the decidability of termination, reachability, coverability and deadlock-freeness of Petri nets endowed with a hierarchy of places, and with inhibitor arcs, reset arcs and transfer arcs that respect this hierarchy. We also investigate what happens when we have a mix of these special arcs, some of which respect the hierarchy, while others do not. We settle the decidability status of the above four problems for all combinations of hierarchy, inhibitor, reset and transfer arcs, except the termination problem for two combinations. For both these combinations, we show that deciding termination is as hard as deciding the positivity problem on linear recurrence sequences -- a long-standing open problem.

We investigate the decidability of the monadic second-order (MSO) theory of the structure (N; <, P1,…,Pk), for various unary predicates P1,…,Pk ⊆ N. We focus in particular on 'arithmetic' predicates arising in the study of linear recurrence sequences, such as fixed-base powers Powk = {kn : n ∈ N}, k-th powers Nk = {nk : n ∈ N}, and the set of terms of the Fibonacci sequence Fib = {0, 1, 2, 3, 5, 8, 13,…} (and similarly for other linear recurrence sequences having a single, non-repeated, dominant characteristic root). We obtain several new unconditional and conditional decidability results, a select sample of which are the following:

The main purpose of this paper is using the combinatorial method, the properties of the power series and characteristic roots to study the computational problem of the symmetric sums of a certain second-order linear recurrence sequences, and obtain some new and interesting identities. These results not only improve on some of the existing results, but are also simpler and more beautiful. Of course, these identities profoundly reveal the regularity of the second-order linear recursive sequence, which can greatly facilitate the calculation of the symmetric sums of the sequences in practice.

We establish consequences of the moving form of Schmidt’s Subspace Theorem. Indeed, we obtain inequalities that bound the logarithmic greatest common divisor of moving multivariable polynomials evaluated at moving S S -unit arguments. In doing so, we complement recent work of Levin. As an additional application, we obtain results that pertain to the greatest common divisor problem for algebraic linear recurrence sequences. These observations are motivated by previous related works of Corvaja-Zannier, Levin, and others.

For a 1-dimensional semi-boundary link, Cochran constructed a sequence of Sato-Levine invariants of successively derived links.This is a linear recurrence sequence and conversely any linear recurrence sequence can be constructed in this way.An upper bound for the growth of this sequence is obtained.

Multivariate cryptography is one of the main candidates to guarantee the security of communication in a post quantum era. While multivariate signature schemes are very fast and require only modest computational resources, the key sizes of such schemes are quite large. In [17] Petzoldt et al. proposed a way to use Linear Recurring Sequences (LRS’s) for the key generation of the Unbalanced Oil and Vinegar (UOV) signature scheme by which they were able to reduce the public key size of this scheme by a factor of 7. In this paper we describe a modification of their scheme, which enables us not only to reduce the public key size, but also to speed up the verification process of the UOV scheme by a factor of 5.KeywordsMultivariate CryptographyUOV Signature SchemeKey Size ReductionFast Verification

Let q = p s be a power of a prime number p and let $${\mathbb {F}_{\rm q}}$$ be a finite field with q elements. This paper aims to demonstrate the utility and relation of composed products to other areas such as the factorization of cyclotomic polynomials, construction of irreducible polynomials, and linear recurrence sequences over $${\mathbb {F}_{\rm q}}$$ . In particular we obtain the explicit factorization of the cyclotomic polynomial $${\Phi_{2^nr}}$$ over $${\mathbb {F}_{\rm q}}$$ where both r ? 3 and q are odd, gcd(q, r) = 1, and $${n\in \mathbb{N}}$$ . Previously, only the special cases when r = 1, 3, 5, had been achieved. For this we make the assumption that the explicit factorization of $${\Phi_r}$$ over $${\mathbb {F}_{\rm q}}$$ is given to us as a known. Let $${n = p_1^{e_1}p_2^{e_2}\cdots p_s^{e_s}}$$ be the factorization of $${n \in \mathbb{N}}$$ into powers of distinct primes p i , 1 ? i ? s. In the case that the multiplicative orders of q modulo all these prime powers $${p_i^{e_i}}$$ are pairwise coprime, we show how to obtain the explicit factors of $${\Phi_{n}}$$ from the factors of each $${\Phi_{p_i^{e_i}}}$$ . We also demonstrate how to obtain the factorization of $${\Phi_{mn}}$$ from the factorization of $${\Phi_n}$$ when q is a primitive root modulo m and $${{\rm gcd}(m, n) = {\rm gcd}(\phi(m),{\rm ord}_n(q)) = 1.}$$ Here $${\phi}$$ is the Euler's totient function, and ord n (q) denotes the multiplicative order of q modulo n. Moreover, we present the construction of a new class of irreducible polynomials over $${\mathbb {F}_{\rm q}}$$ and generalize a result due to Varshamov (Soviet Math Dokl 29:334---336, 1984).

We consider the Diophantine equation $U_n=p^xq^y$, where $U=(U_n)_{n\geq 0}$ is a linear recurrence sequence, $p$ and $q$ are distinct prime numbers and $x,y$ are non-negative integers not both zero. We show that under some technical assumptions the Dioph

We consider the decidability and complexity of the Ultimate Positivity Problem, which asks whether all but finitely many terms of a given rational linear recurrence sequence (LRS) are positive. Using lower bounds in Diophantine approximation concerning sums of S-units, we show that for simple LRS (those whose characteristic polynomial has no repeated roots) the Ultimate Positivity Problem is decidable in polynomial space. If we restrict to simple LRS of a fixed order then we obtain a polynomial-time decision procedure. As a complexity lower bound we show that Ultimate Positivity for simple LRS is at least as hard as the decision problem for the universal theory of the reals: a problem that is known to lie between coNP and PSPACE.

Let (u n ) n be a linear recurrence sequence of integers and let b > 1 be a natural number. In this paper, we show that under some mild technical assumptions the base b expansion of |u n | has at least clog n/log log n non-zero digits when n is large, where c > 0 is a computable constant depending on the initial sequence (u n ) n and b. Our results complement the results of C.L. Stewart from [9]. Some diophantine applications are also presented.

Let \(S= \{ p_1, \ldots, p_s\}\) be a finite, non-empty set of distinct prime numbers and \((U_{n})_{n \geq 0}\) be a linear recurrence sequence of integers of order at least 2. For any positive integer k, and \(w = (w_k, \ldots, w_1)\in\mathbb{Z}^k, w_1, \ldots, w_k\neq 0\) we define \((U_j^{(k, w)})_{j\geq 1}\) an increasing sequence composed of integers of the form \(|w_kU_{n_k} +\cdots + w_1U_{n_1}|\), \( n_k>\cdots >n_1\). Under certain assumptions, we prove that for any \(\varepsilon >0\), there exists an integer \(n_{0}\) such that \([U_j^{(k,w)}]_S < (U_j^{(k, w)})^{\varepsilon},\) \({\rm for}\, j > n_0\), where \([m]_S\) denotes the S-part of the positive integer m. On further assumptions on \((U_{n})_{n \geq 0}\), we also compute an effective bound for \([U_j^{(k, w)}]_S\) of the form \((U_j^{(k,w)})^{1-c}\), where c is a positive constant depending only on r, \(a_1\) , . . ., \(a_r\), \(U_0\), . . ., \(U_{r-1}\) , \(w_1\), . . ., \(w_k\) and S.