Some remarks on a problem of Erdős and Graham (Q1823987)

For each strictly increasing sequence A of natural numbers let \[ P(A)=\{\sum^{\infty}_{k=0}\epsilon_ ka_ k :\quad \epsilon_ k=0\quad or\quad 1,\quad \sum^{\infty}_{k=0}\epsilon_ k<\infty \}. \] Let \(\alpha\), \(\beta\) be real numbers with \(\alpha\) /\(\beta\) irrational. Erdős and Graham conjectured that the sequence \(S=\{[\alpha],[\beta],...,[2^ n\alpha],[2^ n\beta],...\}\) contained all sufficiently large numbers or is said to be complete. This conjecture is reformulated and dyadic representation used to prove the partial form of the conjecture that if in the dyadic representation \(\alpha =\sum^{\infty}_{i=-k}\epsilon_ i2^{-i}\), where \(\epsilon_ i=1\) infinitely often, and if in the dyadic representation for \(\beta\), \(\epsilon_ i=1\) only finitely often, then S is complete. Other results which lend support to the conjecture are proved and stronger versions of the conjecture proposed \((\alpha /\beta \neq 2^{-n})\).