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Miklos Schweitzer 1972_3

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November 5, 2008
limitcalculusintegrationadvanced fieldsadvanced fields unsolved

Problem Statement

Let λi  (i=1,2,...) \lambda_i \;(i=1,2,...) be a sequence of distinct positive numbers tending to infinity. Consider the set of all numbers representable in the form μ=∑i=1∞niλi, \mu= \sum_{i=1}^{\infty}n_i\lambda_i , where ni≥0 n_i \geq 0 are integers and all but finitely many ni n_i are 0 0. Let L(x)=∑λi≤x1  and   M(x)=∑μ≤x1 . L(x)= \sum _{\lambda_i \leq x} 1 \;\textrm{and}\ \;M(x)= \sum _{\mu \leq x} 1 \ . (In the latter sum, each μ \mu occurs as many times as its number of representations in the above form.) Prove that if lim⁡x→∞L(x+1)L(x)=1, \lim_{x\rightarrow \infty} \frac{L(x+1)}{L(x)}=1, then lim⁡x→∞M(x+1)M(x)=1. \lim_{x\rightarrow \infty} \frac{M(x+1)}{M(x)}=1. G. Halasz
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