OBJECTIVE AND REVIEW OF ALL VERSIONS
Erdős #260; independent attempt, 2026-09-23. For increasing integers $a_n$ with $a_n/n\to\infty$, must $\sum a_n2^{-a_n}$ be irrational? I read
260.tex and
260_v2.tex; both contain the same erroneous substitution in the tail estimate. Their large-gap conclusion can be repaired by a shorter denominator proof.
WORK: CORRECTED TAIL AND RATIONALITY OBSTRUCTION
Write $P_n=\sum_{i\le n}a_i2^{-a_i}$ and $T_n=S-P_n$. If $S=p/q\in\mathbb Q$, positivity of the infinite tail and the denominator of $P_n$ imply \[
T_n\ge\frac1{q2^{a_n}}.
\] The correct geometric-derivative identity is \[
\sum_{m=A}^\infty\frac m{2^m}=\frac{2(A+1)}{2^A}.
\] The source instead substitutes $(A+2)/2^A$, which omits the first term by shifting the summation endpoint. Distinctness of $a_i$ gives the valid bound \[
T_n\le\frac{2(a_{n+1}+1)}{2^{a_{n+1}}}.
\] Combining these bounds forces \[
a_{n+1}-a_n-\log_2(a_{n+1}+1)\le\log_2(2q)
\quad\text{for every }n.
\] Therefore an unbounded limsup on the left proves irrationality, recovering the intended logarithmic-gap criterion without any claim about frozen binary digits or carrying.
ADVERSARIAL VERIFICATION AND REMAINING GAP
The argument covers dyadic and non-dyadic rationals simultaneously; the positive remainder cannot be zero. It does not follow that the hypothesis $a_n/n\to\infty$ forces this logarithmic-gap limsup. For instance $a_n=\lceil n\sqrt{\log(n+2)}\rceil$ grows faster than $n$ but has gaps smaller than $\log a_n$. Its irrationality is known by stronger results on gaps tending to infinity, but those results do not handle all sequences in the original question. This attempt corrects the supplied proof, not the general conjecture, and claims no improved known criterion.
SOURCES CHECKED
Both local versions listed above; official indexed statement and existing sufficient conditions checked at
https://www.erdosproblems.com/260, 2026-09-23.
FINAL
UNRESOLVED. The missing factor in the tail bound is repaired; the full density-zero exponent-set hypothesis remains insufficient for this argument.
COMPLETION ESTIMATE
Conservative subjective progress toward the original growth hypothesis.
COMPLETION: 10%