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A generalization of a result on integers in metacyclic extensions

James Carter (1999)

Colloquium Mathematicae

Let p be an odd prime and let c be an integer such that c>1 and c divides p-1. Let G be a metacyclic group of order pc and let k be a field such that pc is prime to the characteristic of k. Assume that k contains a primitive pcth root of unity. We first characterize the normal extensions L/k with Galois group isomorphic to G when p and c satisfy a certain condition. Then we apply our characterization to the case in which k is an algebraic number field with ring of integers ℴ, and, assuming some...

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 ℕ * . 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 ∈ ℕ * 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.

A generalization of a theorem of Schinzel

Georges Rhin (2004)

Colloquium Mathematicae

We give lower bounds for the Mahler measure of totally positive algebraic integers. These bounds depend on the degree and the discriminant. Our results improve earlier ones due to A. Schinzel. The proof uses an explicit auxiliary function in two variables.

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