THE FIRST JONES PROBLEM A CANTOR SET OF POSITIVE MEASURE

This paper resolves the first Jones problem on achievement sets. We prove that the set of all subsums of the series with terms 2⁻ⁿ + (−1)ⁿ3⁻ⁿ is a Cantor set of positive Lebesgue measure. More generally, the same conclusion holds for the family aₙ(t) = 2⁻ⁿ + t(−1)ⁿ3⁻ⁿ for every nonzero real parameter t, whereas the unperturbed case t = 0 gives the interval [0, 1]. The proof combines cylinder decompositions, tail rescaling, measure estimates, and typical binary expansions. Its central argument is an induction connecting interval coverings by finitely many translates with the multiplicity of representations at interior points. An integral identity establishes that this multiplicity is finite almost everywhere; together with the induction, this excludes nonempty interior. The paper also provides explicit lower bounds for Lebesgue measure, including a bound of 1/4 for Jones’s original set, and proves that the largest connected components of the standard finite outer approximations tend to zero in length. An extension of the method establishes the same topological classification for a companion perturbed binary sequence discussed in the survey of Głąb and Prus-Wiśniowski.