Deletion thresholds and exponential examples for complete sequences

Abstract

We prove that the pairs of integers $0\le m 0$ 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.