Lacunary sequences whose reciprocal sums represent all rational numbers in an interval
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909940924284928 |
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| author | van Doorn, Wouter Kovač, Vjekoslav |
| author_facet | van Doorn, Wouter Kovač, Vjekoslav |
| contents | Disproving a conjecture of Bleicher and Erdős, we show that there exists a lacunary sequence of positive integers such that finite sums of reciprocals of its terms attain all rational numbers from a non-empty open interval. We also study several stronger variants of their original problem: determining the value of the optimal lacunarity parameter, representing rational numbers infinitely many times, finding such lacunary sequences with arbitrarily large jumps, and relating the maximal length of a filled interval to a prescribed lacunarity parameter. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_24971 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Lacunary sequences whose reciprocal sums represent all rational numbers in an interval van Doorn, Wouter Kovač, Vjekoslav Number Theory Classical Analysis and ODEs Combinatorics 11D68, 11B05, 11B13, 40A05 Disproving a conjecture of Bleicher and Erdős, we show that there exists a lacunary sequence of positive integers such that finite sums of reciprocals of its terms attain all rational numbers from a non-empty open interval. We also study several stronger variants of their original problem: determining the value of the optimal lacunarity parameter, representing rational numbers infinitely many times, finding such lacunary sequences with arbitrarily large jumps, and relating the maximal length of a filled interval to a prescribed lacunarity parameter. |
| title | Lacunary sequences whose reciprocal sums represent all rational numbers in an interval |
| topic | Number Theory Classical Analysis and ODEs Combinatorics 11D68, 11B05, 11B13, 40A05 |
| url | https://arxiv.org/abs/2509.24971 |