A series with a Cantorval set of subsums and ratio of consecutive terms tending to 1/2 (the second Jones problem)
Abstract
The second Jones problem (2011) consists in determining whether there exists a convergent positive series for which the ratio of consecutive terms has a limit and the set of subsums is a Cantorval; in the survey by Gł{\k a}b and Prus-Wiśniowski (2026) this problem is included in the list of open problems. In this paper, a family of such series is constructed explicitly as combinations of blocks of the form (2k−1)/2t + (2k−1)/2t + (2k−2)/2t + ⋯ + 2/2t, where t is the index of the last term of the block in the series and k≥2 is the length of the block. For each series of the family it is proved that its set of subsums contains an interval and is not a finite union of closed intervals, and hence is a Cantorval. The proof relies on mixed-radix numerical systems with a redundant alphabet and uses an idea of Guthrie and Nymann (1988). If the lengths of the blocks tend to infinity, then the limit of the ratio of consecutive terms of such series is 1/2, which, by a result of Jones, is the only possible value of the limit for which the set of subsums of a series can be a Cantorval. Thus, an affirmative answer to the second Jones problem is obtained.
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BibTeX
@misc{pratsiovytyi2026series,
title = {{A series with a Cantorval set of subsums and ratio of consecutive terms tending to $1/2$ (the second Jones problem)}},
author = {Mykola Pratsiovytyi and Yaroslav Vynnyshyn and Oleh Vynnyshyn},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.14251},
eprint = {2610.14251},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.14251},
}
RIS
TY - UNPB
TI - A series with a Cantorval set of subsums and ratio of consecutive terms tending to 1/2 (the second Jones problem)
AU - Mykola Pratsiovytyi
AU - Yaroslav Vynnyshyn
AU - Oleh Vynnyshyn
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.14251
UR - https://arxiv.org/abs/2610.14251
AB - The second Jones problem (2011) consists in determining whether there exists a convergent positive series for which the ratio of consecutive terms has a limit and the set of subsums is a Cantorval; in the survey by Gł{\k a}b and Prus-Wiśniowski (2026) this problem is included in the list of open problems. In this paper, a family of such series is constructed explicitly as combinations of blocks of the form (2^k−1)/2^t + (2^k−1)/2^t + (2^k−2)/2^t + ⋯ + 2/2^t, where t is the index of the last term of the block in the series and k≥2 is the length of the block. For each series of the family it is proved that its set of subsums contains an interval and is not a finite union of closed intervals, and hence is a Cantorval. The proof relies on mixed-radix numerical systems with a redundant alphabet and uses an idea of Guthrie and Nymann (1988). If the lengths of the blocks tend to infinity, then the limit of the ratio of consecutive terms of such series is 1/2, which, by a result of Jones, is the only possible value of the limit for which the set of subsums of a series can be a Cantorval. Thus, an affirmative answer to the second Jones problem is obtained.
ER -