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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.

On B 2 k -sequences

Martin Helm — 1993

Acta Arithmetica

Introduction. An old conjecture of P. Erdős repeated many times with a prize offer states that the counting function A(n) of a B r -sequence A satisfies l i m i n f n ( A ( n ) / ( n 1 / r ) ) = 0 . The conjecture was proved for r=2 by P. Erdős himself (see [5]) and in the cases r=4 and r=6 by J. C. M. Nash in [4] and by Xing-De Jia in [2] respectively. A very interesting proof of the conjecture in the case of all even r=2k by Xing-De Jia is to appear in the Journal of Number Theory [3]. Here we present a different, very short proof of Erdős’...

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