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0 - 1 sequences having the same numbers of ( 1 - 1 ) -couples of given distances

Antonín Lešanovský, Jan Rataj, Stanislav Hojek (1992)

Mathematica Bohemica

Let a be a 0 - 1 sequence with a finite number of terms equal to 1. The distance sequence δ ( a ) of a is defined as a sequence of the numbers of ( 1 - 1 ) -couples of given distances. The paper investigates such pairs of 0 - 1 sequences a , b that a is different from b and δ ( a ) = δ ( b ) .

A contribution to infinite disjoint covering systems

János Barát, Péter P. Varjú (2005)

Journal de Théorie des Nombres de Bordeaux

Let the collection of arithmetic sequences { d i n + b i : n } i I be a disjoint covering system of the integers. We prove that if d i = p k q l for some primes p , q and integers k , l 0 , then there is a j i such that d i | d j . We conjecture that the divisibility result holds for all moduli.A disjoint covering system is called saturated if the sum of the reciprocals of the moduli is equal to 1 . The above conjecture holds for saturated systems with d i such that the product of its prime factors is at most 1254 .

A function related to the central limit theorem

Paul Bracken (1998)

Commentationes Mathematicae Universitatis Carolinae

A number of properties of a function which originally appeared in a problem proposed by Ramanujan are presented. Several equivalent representations of the function are derived. These can be used to evaluate the function. A new derivation of an expansion in inverse powers of the argument of the function is obtained, as well as rational expressions for higher order coefficients.

A generalization of a theorem of Erdös on asymptotic basis of order 2

Martin Helm (1994)

Journal de théorie des nombres de Bordeaux

Let 𝒯 be a system of disjoint subsets of * . In this paper we examine the existence of an increasing sequence of natural numbers, A , that is an asymptotic basis of all infinite elements T j of 𝒯 simultaneously, satisfying certain conditions on the rate of growth of the number of representations 𝑟 𝑛 ( 𝐴 ) ; 𝑟 𝑛 ( 𝐴 ) : = ( 𝑎 𝑖 , 𝑎 𝑗 ) : 𝑎 𝑖 < 𝑎 𝑗 ; 𝑎 𝑖 , 𝑎 𝑗 𝐴 ; 𝑛 = 𝑎 𝑖 + 𝑎 𝑗 , for all sufficiently large n T j and j * A theorem of P. Erdös is generalized.

A generalization of a theorem of Erdős-Rényi to m-fold sums and differences

Kathryn E. Hare, Shuntaro Yamagishi (2014)

Acta Arithmetica

Let m ≥ 2 be a positive integer. Given a set E(ω) ⊆ ℕ we define r N ( m ) ( ω ) to be the number of ways to represent N ∈ ℤ as a combination of sums and differences of m distinct elements of E(ω). In this paper, we prove the existence of a “thick” set E(ω) and a positive constant K such that r N ( m ) ( ω ) < K for all N ∈ ℤ. This is a generalization of a known theorem by Erdős and Rényi. We also apply our results to harmonic analysis, where we prove the existence of certain thin sets.

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