Discrepancy of Arithmetic Progressions in Boxes and Convex Bodies
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| Format: | Preprint |
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2025
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| _version_ | 1866914278899974144 |
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| author | Li, Lily Nikolov, Aleksandar |
| author_facet | Li, Lily Nikolov, Aleksandar |
| contents | The combinatorial discrepancy of arithmetic progressions inside $[N] := \{1, \ldots, N\}$ is the smallest integer $D$ for which $[N]$ can be colored with two colors so that any arithmetic progression in $[N]$ contains at most $D$ more elements from one color class than the other. Bounding the discrepancy of such set systems is a classical problem in discrepancy theory. More recently, this problem was generalized to arithmetic progressions in grids like $[N]^d$ (Valk{ó}) and $[N_1]\times \ldots \times [N_d]$ (Fox, Xu, and Zhou). In the latter setting, Fox, Xu, and Zhou gave upper and lower bounds on the discrepancy that match within a $\frac{\log |Ω|}{\log \log |Ω|}$ factor, where $Ω:= [N_1]\times \ldots \times [N_d]$ is the ground set. In this work, we use the connection between factorization norms and discrepancy to improve their upper bound to be within a $\sqrt{\log|Ω|}$ factor from the lower bound. We also generalize Fox, Xu, and Zhou's lower bound, and our upper bounds to arithmetic progressions in arbitrary convex bodies. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_12598 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Discrepancy of Arithmetic Progressions in Boxes and Convex Bodies Li, Lily Nikolov, Aleksandar Combinatorics Discrete Mathematics 11K38 (Primary) 11B25 (Secondary) The combinatorial discrepancy of arithmetic progressions inside $[N] := \{1, \ldots, N\}$ is the smallest integer $D$ for which $[N]$ can be colored with two colors so that any arithmetic progression in $[N]$ contains at most $D$ more elements from one color class than the other. Bounding the discrepancy of such set systems is a classical problem in discrepancy theory. More recently, this problem was generalized to arithmetic progressions in grids like $[N]^d$ (Valk{ó}) and $[N_1]\times \ldots \times [N_d]$ (Fox, Xu, and Zhou). In the latter setting, Fox, Xu, and Zhou gave upper and lower bounds on the discrepancy that match within a $\frac{\log |Ω|}{\log \log |Ω|}$ factor, where $Ω:= [N_1]\times \ldots \times [N_d]$ is the ground set. In this work, we use the connection between factorization norms and discrepancy to improve their upper bound to be within a $\sqrt{\log|Ω|}$ factor from the lower bound. We also generalize Fox, Xu, and Zhou's lower bound, and our upper bounds to arithmetic progressions in arbitrary convex bodies. |
| title | Discrepancy of Arithmetic Progressions in Boxes and Convex Bodies |
| topic | Combinatorics Discrete Mathematics 11K38 (Primary) 11B25 (Secondary) |
| url | https://arxiv.org/abs/2504.12598 |