Reading Eight Erdős Problems Together

What was already known

• The covering argument of Theorem 6.1(ii) goes back to Kakeya; see [9][10][11]. Its use to build rational series inside a class defined by soft data is the method of Kovač and Tao [13], of Crmarić and Kovač [12] and of van Doorn and Kovač [38].

• That the subsums of $\sum (t^n-1{)}^{-1}$ form a Cantor set at fixed $t\ge 2$ is [13].

• A closed set of positive measure can contain essentially no rationals [26], so the heuristic of Section 6.4 cannot be a consequence of measure.

• The rational-point counting papers examined here concern null Cantor sets [27][28]. We did not locate a theorem settling the present positive-measure subsum problem. The searches were made on 20 September 2026 and are listed in the repository record.

• One identity with a consequence. Since $\sum _{n\ge 1}\mu (n)/(b^n-1)=1/b$, the sums of $(b^n-1{)}^{-1}$ over squarefree $n$ with an even, respectively odd, number of prime factors are $\frac{1}{2}(X_{\mathrm{sf}}(b)\pm 1/b)$. Duverney and Tachiya prove that $X_{\mathrm{sf}}(2^j)$ is irrational [39], as quoted in [33], so both sums are irrational at every base $2^j$. Both supports have divergent reciprocal sums. The identity at base $2$ is derived in [31]; the corollary may be known.