Geodesic
Exponential odd-distance sets under the Manhattan metric
Alberto Espuny Díaz1 ; Emma Hogan2 ; Freddie Illingworth3 ; Lukas Michel2 ; Julien Portier4 ; Jun Yan2
1Universitat de Barcelona
2University of Oxford
3University College London
4Ecole Polytechnique Federale de Lausanne
The electronic journal of combinatorics, Tome 32 (2025) no. 4
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We construct a set of $2^n$ points in $\mathbb{R}^n$ such that all pairwise Manhattan distances are odd integers, which improves the recent linear lower bound of Golovanov, Kupavskii and Sagdeev. In contrast to the Euclidean and maximum metrics, this shows that the odd-distance set problem behaves very differently to the equilateral set problem under the Manhattan metric. Moreover, all coordinates of the points in our construction are integers or half-integers, and we show that our construction is optimal under this additional restriction.
DOI : 10.37236/13530
Classification : 52C10, 05D99, 05C35, 46B20, 52C30
Mots-clés : odd-distance set problem, Manhattan metric

Alberto Espuny Díaz  1   ; Emma Hogan  2   ; Freddie Illingworth  3   ; Lukas Michel  2   ; Julien Portier  4   ; Jun Yan  2

1 Universitat de Barcelona
2 University of Oxford
3 University College London
4 Ecole Polytechnique Federale de Lausanne 
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     author = {Alberto Espuny D{\'\i}az and Emma Hogan and Freddie Illingworth and Lukas Michel and Julien Portier and Jun Yan},
     title = {Exponential odd-distance sets under the {Manhattan} metric},
     journal = {The electronic journal of combinatorics},
     year = {2025},
     volume = {32},
     number = {4},
     doi = {10.37236/13530},
     zbl = {8120118},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/13530/}
}
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Alberto Espuny Díaz; Emma Hogan; Freddie Illingworth; Lukas Michel; Julien Portier; Jun Yan. Exponential odd-distance sets under the Manhattan metric. The electronic journal of combinatorics, Tome 32 (2025) no. 4. doi: 10.37236/13530

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