Reciprocal Mersenne Subseries

Two proof details and sources

The comparison of displacement atoms works for real bases: ${\mathrm{\Delta }}_{b,A}(N)\le {\mathrm{\Delta }}_{2,A}(N)$ whenever $b\ge 2$. To see this, cancel $b-1$ and differentiate $\sum _{i<r}{b}^i/\sum _{i<d}{b}^i$. The numerator of the derivative is $\sum _{i<r\le j<d}(i-j)b^{i+j-1}\le 0$. An integer base is needed only when the displacement is placed on a rational lattice.

For the scalar positive-cover cost, put $z={\log }_{2}t$. Direct differentiation of $y^z/(y-1)$ on $1<y\le 2$ gives

These two calculations supply the elementary details used in the short paper.

The original periodic and linear-independence papers of Luca and Tachiya are now distinguished from their later overview [14][15][3]. Their theorem formulations used here were read in the author-written overview; the two original publisher PDFs were not retrieved. Van Assche’s nonvanishing and small-linear-form argument for the full Lambert series [16] is a comparator for the missing upper-height control, not a result for arbitrary thinnings.

For achievement-set background, Hornich’s original bibliographic record [17] is separated from the exposition actually used here [4]. Głąb and Prus-Wiśniowski’s survey [18] provides a modern map of geometry and cardinality questions, not a rational-membership theorem for the present set.