Reading Eight Erdős Problems Together

Further questions

1. Is $1/2$ a subsum of $\sum (2^n-1{)}^{-1}$? The exact obligation is in [33]. By Lemma 6.8 the general question is one-sided approximation of a rational by finite subsums $X_F$ to within $g_{maxF}$.

2. Does the count of fractions of height at most $Q$ rejected at step $n$ stay close to $(3Q^2/{\pi }^2)2^{n-1}{g}_n$ in the joint range $n\le (2-\epsilon ){\log }_{2}Q$, counted modulo the translations of Lemma 6.9? A persistent excess would be the first sign of an arithmetic mechanism for #257.

3. Theorem 6.6 settles every divisibility chain at every real algebraic base $t>1$, and includes non-chain hosts of unbounded width. Which supports lacking separated cuts admit comparable control of the cleared prefix height and the initial tail patterns? The full support at $3/2$ still has neither conclusion nor such a transfer here.

4. Is there one inequality behind #1049 Theorem 5 and the cover cost of #257?

The capacity criterion already covers non-power and oscillating allowances. For factorial gaps of fixed length $m$, a bounded multiple of $n^m$ still gives the lattice obstruction, whereas $n^{m}L(n)$ with $L(n)\to \mathrm{\infty }$ permits interval filling. A further question concerns the null case: what finer tail data determine its Hausdorff dimension? The criterion itself does not separate dimension zero from full-dimensional null sets. Outside integer divisibility chains the prefix lattice changes, so no corresponding necessity is asserted here.