Zeros of linear recurrence sequences

Let be an exponential polynomial over a field of zero characteristic. Assume that for each pair i,j with i≠j, α _i /α _j is not a root of unity. Define . We introduce a partition of into subsets (1≤i≤m), which induces a decomposition of f into , so that, for 1≤i≤m, , while for , the number either is transcendental or else is algebraic with not too small a height. Then we show that for all but at most solutions x∈ℤ of f(x)= 0, we have

Palindromes in linear recurrence sequences

We prove that for any base $b\ge 2$ and for any linear homogeneous recurrence sequence $\{a_n\}_{n\ge 1}$ satisfying certain conditions, there exits a positive constant $c>0$ such that $\# \{n\le x:\ a_n \;\text{ is} \text{ palindromic} \text{ in} \text{ base}\; b\} \ll x^{1-c}$ .     

On the nonlinearity of linear recurrence sequences

We obtain an upper bound on exponential sums of a new type with linear recurrence sequences. We apply this bound to estimate the Fourier coefficients, and thus the nonlinearity, of a Boolean function associated with a linear recurrence sequence in a natural way.     

Positivity of third order linear recurrence sequences

It is shown that the Positivity Problem for a sequence satisfying a third order linear recurrence with integer coefficients, i.e., the problem whether each element of this sequence is nonnegative, is decidable.     

On the multiplicity of linear recurrence sequences

We prove a lemma regarding the linear independence of certain vectors and use it to improve on a bound due to Schmidt on the zero-multiplicity of linear recurrence sequences.     

Sums of primes and quadratic linear recurrence sequences

Let u be a sequence of positive integers which grows essentially as a geometric progression. We give a criterion on u in terms of its distribution modulo d, d = 1, 2,..., under which the set of positive integers expressible by the sum of a prime number and an element of u has a positive lower density. This criterion is then checked for some second order linear recurrence sequences. It follows, for instance, that the set of positive integers of the form p ＋ [（2 ＋ √3）n], where p is a prime number and n is a positive integer, has a positive lower density. This generalizes a recent result of Enoch Lee. In passing, we show that the periods of linear recurrence sequences of order m modulo a prime number p cannot be ＂too small＂ for most prime numbers p.     

Distinct digits in base b expansions of linear recurrence sequences

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.     

Arithmetic linear series with base conditions

In this note, we study the volume of arithmetic linear series with base conditions. As an application, we consider the problem of Zariski decompositions on arithmetic varieties.     

Approximation of analytical functions by sequences of k-positive linear operators

Approximation properties of sequences of k-positive operators, i.e. linear operators acting in the space of analytical functions and preserving the cone of functions with non-negative Taylor coefficients, are studied. Some general theorems which are valid in the space of functions that are analytical in a bounded simply connected domain are proved.     

Weighted approximation of continuous functions by sequences of linear positive operators

In this work we obtain, under suitable conditions, theorems of Korovkin type for spaces with different weight, composed of continuous functions defined on unbounded regions. These results can be seen as an extension of theorems by Gadjiev in [4] and [5].     

Arithmetic properties of functions associated with Dirichlet L-functions

This article considers functions of the form , where is the Jacobian quadratic symbol, d runs through all natural divisors of a given number r, j = 1, ..., s. Linear independence is proved over the field of rationals for the values of these functions on small rational x. Effective lower bounds are obtained for linear forms with rational integral coefficients. The results, in particular, strengthen known bounds for polylogarithms. Hermite-Padé approximations of the second kind are used.Translated from Matematicheskie Zametki, Vol. 52, No. 1, pp. 120–127, July, 1992.     

On the residue classes of integer sequences satisfying a linear three term recurrence formula

In this paper we investigate linear three-term recurrence formulae with sequences of integers (T(n))_n?0 and (U(n))_n?0, which are ultimately periodic modulo m, e.g.

     

Rational functions with linear relations

We find all polynomials over a field such that and are linear and . We also solve the same problem for rational functions , in case the field is algebraically closed.

     

Arithmetic of Levitan's generalized characteristic functions

Some properties are stated for functions of the form, where is a distribution on the halfline [0,).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 61, pp. 56–58, 1976.I thank Prof. I. V. Ostrovskii for suggesting this problem and for his support.     

Arithmetic functions and the Cauchy product

It is known that under the Dirichlet product, the set of arithmetic functions in several variables becomes a unique factorization domain. A. Zaharescu and M. Zaki proved an analog of the ABC conjecture in this ring and showed that there exists a non-trivial solution to the Fermat equation $$z^n=x^n+y^n$$ ($$nge 3$$). It is also known that under the Cauchy product, the set of arithmetic functions becomes a unique factorization domain. In this paper, we consider the ring of arithmetic functions in several variables under the Cauchy product and prove an analog of the ABC conjecture in this ring to show that there exists a non-trivial solution to the Fermat equation $$z^n=x^n+y^n$$ ($$nge 3$$).