Discrepancy of Arithmetic Progressions in Boxes and Convex Bodies
Lily Li, Aleksandar Nikolov
TL;DR
The paper tackles the discrepancy of arithmetic progressions in high-dimensional grids and convex bodies by linking discrepancy to the γ2 factorization norm. It proves a tight upper bound disc(𝒜_N) ≲_d √{log|Ω|} · f(N) for APs in axis-aligned boxes and extends analogous results to general convex bodies, using a prefix-discrepancy framework and γ2-based constructive coloring (where available). A matching lower bound is established via Fourier-analytic methods, showing that the growth rate in f(K) is intrinsic, with additional corollaries for scaling in convex bodies. The results broaden the scope of discrepancy bounds beyond boxes to arbitrary convex bodies, and provide both non-constructive and constructive tools depending on the geometric setting.
Abstract
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.