Advanced Search

Let $𝒯$ be a system of disjoint subsets of ${\mathbb{N}}^{*}$. 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 ${𝑟}_{𝑛}\left(𝐴\right);{𝑟}_{𝑛}\left(𝐴\right):=\left|\left\{({𝑎}_{𝑖},{𝑎}_{𝑗}):{𝑎}_{𝑖}\<{𝑎}_{𝑗};{𝑎}_{𝑖},{𝑎}_{𝑗}\in 𝐴;𝑛={𝑎}_{𝑖}+{𝑎}_{𝑗}\right\}\right|$, for all sufficiently large $n\in T_j$ and $j\in {\mathbb{N}}^{*}$ A theorem of P. Erdös is generalized.

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 $limin{f}_{n\to \infty }(A\left(n\right)/\left(n^{1/r}\right))=0$. 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’...