完整序列的删除阈值与指数型例子
Deletion thresholds and exponential examples for complete sequences
AI总结：
本文解决了Erdős问题348，刻画了完整序列的删除阈值，并构造反例否证了Graham关于指数序列完整性的猜想。
AI中文摘要：
我们证明了对于非降整数序列，在每次删除$m$项后仍保持完整、而在每次删除$n$项后变得不完整的整数对$0\le m<n$恰好是满足$m\le1$的那些。这里，一个序列是完整的，如果每个足够大的整数都可以表示为该序列中具有不同下标的项的有限和。这回答了由Graham提出、后来收录于Erdős和Graham合著书籍中的Erdős问题348。证明使用了一个中心区间定理：若一个完整的非降正整数序列$(a_i)$的前缀和$S_j$满足$S_j-a_{j+1}\to\infty$，则每个足够长的前缀都能表示从任意固定完整性阈值$T$到$S_j-T$之间的所有整数。我们还否证了Graham的猜想（后来Erdős和Graham也重复过），即对于每个$t>0$和$1<\gamma<(1+\sqrt5)/2$，序列$(\lfloor t\gamma^n\rfloor)_{n\ge1}$是完整的。我们通过结合Dubickas的分数部分定理与一个初等的符号调整来获得反例。对于该范围内的一个公共底数，我们进一步构造了两个这样的序列，它们的交错是不完整的，并且其系数之比不是底数的任何整数次幂的有理倍数。
英文摘要：
We prove that the pairs of integers $0\le m<n$ for which a nondecreasing integer sequence can remain complete after every deletion of $m$ terms and become incomplete after every deletion of $n$ terms are exactly those with $m\le1$. Here a sequence is complete if every sufficiently large integer is a finite sum of terms with distinct indices. This answers Erdős Problem 348, posed by Graham and later included in the book of Erdős and Graham. The proof uses a central-interval theorem: if a complete nondecreasing positive integer sequence $(a_i)$ has prefix sums $S_j$ with $S_j-a_{j+1}\to\infty$, then each sufficiently long prefix represents every integer from any fixed completeness threshold $T$ to $S_j-T$. We also refute Graham's conjecture, later repeated by Erdős and Graham, that $(\lfloor tγ^n\rfloor)_{n\ge1}$ is complete for every $t>0$ and $1<γ<(1+\sqrt5)/2$. We obtain the counterexample by combining Dubickas's fractional-part theorem with an elementary sign adjustment. For a common base in this range, we further construct two such sequences whose interleaving is incomplete and whose coefficient ratio is not a rational multiple of any integer power of the base.
