Combinatorial designs and the Prouhet--Tarry--Escott problem
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
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2026
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| _version_ | 1866917352199684096 |
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| author | Inagaki, Munenori Matsumura, Hideki Sawa, Masanori Uchida, Yukihiro |
| author_facet | Inagaki, Munenori Matsumura, Hideki Sawa, Masanori Uchida, Yukihiro |
| contents | This is the first paper that provides a systematic treatment of the $r$-dimensional PTE problem in additive number theory, abbreviated by PTE$_r$, through its connection with combinatorial design theory, the branch of combinatorial mathematics that deals with finite set systems or arrangements with the ^^ balancedness' conditions. We first propose a combinatorial reconsideration of the definition of nontrivial solution introduced by Alpers and Tijdeman (2007), and then prove a fundamental lower bound for the size of such solutions. We exhibit high-dimensional minimal solutions with respect to the fundamental bound, which inherently have the structure of distinctive block designs or orthogonal arrays (OAs). Next, we develop a powerful method for constructing PTE$_r$ solutions via various classes of combinatorial designs such as block designs and OAs. Furthermore, we explore two dimension-lifting methods for constructing PTE$_r$ solutions: one is a combinatorial composition that produces PTE$_r$ solutions by embedding lower-dimensional solutions into OAs with $r$ columns, and the other is a recursive technique in which a PET$_r$ solution is constructed by taking the Cartesian product of two lower-dimensional solutions. It is emphasized that our results generalize many previous works, including a measure-theoretic construction by Lorentz (1949) and its geometric analog by Alpers and Tijdeman (2007), a key lemma in Jacroux's work (1995) on the construction of sets of integers with equal power sums, and the famous Borwein solution and its two-dimensional extension by Matsumura and Sawa (2025). In addition, we prove a characterization theorem for ideal solutions of the PTE$_1$ and discuss the connection with a curious phenomenon, called half-integer design, that is rarely reported in the combinatorial design theory or spherical design theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_11100 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Combinatorial designs and the Prouhet--Tarry--Escott problem Inagaki, Munenori Matsumura, Hideki Sawa, Masanori Uchida, Yukihiro Combinatorics Number Theory 11D72, 05B05 (Primary), 05E99, 14G05, 51E15 (Secondary) This is the first paper that provides a systematic treatment of the $r$-dimensional PTE problem in additive number theory, abbreviated by PTE$_r$, through its connection with combinatorial design theory, the branch of combinatorial mathematics that deals with finite set systems or arrangements with the ^^ balancedness' conditions. We first propose a combinatorial reconsideration of the definition of nontrivial solution introduced by Alpers and Tijdeman (2007), and then prove a fundamental lower bound for the size of such solutions. We exhibit high-dimensional minimal solutions with respect to the fundamental bound, which inherently have the structure of distinctive block designs or orthogonal arrays (OAs). Next, we develop a powerful method for constructing PTE$_r$ solutions via various classes of combinatorial designs such as block designs and OAs. Furthermore, we explore two dimension-lifting methods for constructing PTE$_r$ solutions: one is a combinatorial composition that produces PTE$_r$ solutions by embedding lower-dimensional solutions into OAs with $r$ columns, and the other is a recursive technique in which a PET$_r$ solution is constructed by taking the Cartesian product of two lower-dimensional solutions. It is emphasized that our results generalize many previous works, including a measure-theoretic construction by Lorentz (1949) and its geometric analog by Alpers and Tijdeman (2007), a key lemma in Jacroux's work (1995) on the construction of sets of integers with equal power sums, and the famous Borwein solution and its two-dimensional extension by Matsumura and Sawa (2025). In addition, we prove a characterization theorem for ideal solutions of the PTE$_1$ and discuss the connection with a curious phenomenon, called half-integer design, that is rarely reported in the combinatorial design theory or spherical design theory. |
| title | Combinatorial designs and the Prouhet--Tarry--Escott problem |
| topic | Combinatorics Number Theory 11D72, 05B05 (Primary), 05E99, 14G05, 51E15 (Secondary) |
| url | https://arxiv.org/abs/2603.11100 |