On the subset sums of exponential type sequences
Volume 173 / 2016
Abstract
For a sequence $A\subseteq \mathbb{N}$, let $P(A)$ be the set of all sums of distinct terms taken from $A$. The sequence $A$ is said to be \lt i \gt complete \lt /i \gt if $P(A)$ contains all sufficiently large integers. Let $p \gt 1$ be an integer. The following main results are proved: (a) Let $A_t=\{ a_1\le \dots \le a_t\}$ be any sequence of positive integers (not necessarily distinct), $S_p=\{ p^i : i=0, 1, \dots \} $ and $S_p A_t=\{ p^i a_j : i=0, 1, \dots; \, j=1, \dots , t\}$. When $t\ge p-1$, the sequence $P(S_pA_t)$ has positive lower asymptotic density not less than $1/a_{p-1}$. The lower bounds $p-1$ and $1/a_{p-1}$ are both the best possible. (b) For any positive integer $k$, the sequence $\{ p^i F_j : i=0, 1, \dots ;\, j=k, k+1, \dots , n\}$ is complete, where $F_j$ is the $j$th Fibonacci number and $n=p^2 F_{k+2p-1}^2$.
