Recovering affine-linearity of functions from their restrictions to affine lines

Abstract

Motivated by recent results of Tao-Ziegler [Discrete Anal. 2016] and Greenfeld-Tao (2022 preprint) on concatenating affine-linear functions along subgroups of an abelian group, we show three results on recovering affine-linearity of functions $f:V\to W$ from their restrictions to affine lines, where $V,W$ are $\mathbb{F}$-vector spaces and $\dim V⩾2$. First, if $\dim V<|\mathbb{F}|$ and $f:V\to \mathbb{F}$ is affine-linear when restricted to affine lines parallel to a basis and to certain "generic" lines through $0$, then $f$ is affine-linear on $V$. (This extends to all modules $M$ over unital commutative rings $R$ with large enough characteristic.) Second, we explain how a classical result attributed to von Staudt (1850s) extends beyond bijections: if $f:V\to W$ preserves affine lines $\ell$, and if $f(v)\notin f(\ell )$ whenever $v\notin \ell$, then this also suffices to recover affine-linearity on $V$, but up to a field automorphism. In particular, if $\mathbb{F}$ is a prime field $\mathbb{Z}/p\mathbb{Z}$ ($p>2$) or $\mathbb{Q}$, or a completion ${\mathbb{Q}}_p$ or $\mathbb{R}$, then $f$ is affine-linear on $V$. We then quantitatively refine our first result above, via a weak multiplicative variant of the additive $B_h$-sets initially explored by Singer [Trans. Amer. Math. Soc. 1938], Erdos-Turan [J. London Math. Soc. 1941], and Bose-Chowla [Comment. Math. Helv. 1962]. Weak multiplicative $B_h$-sets occur inside all rings with large enough characteristic, and in all infinite or large enough finite integral domains/fields. We show that if $R$ is among any of these classes of rings, and $M=R^n$ for some $n⩾3$, then one requires affine-linearity on at least $\left(\genfrac{}{}{0ex}{}{n}{\lceil n/2\rceil }\right)$-many generic lines to deduce the global affine-linearity of $f$ on $R^n$. Moreover, this bound is sharp.

Minor edits. Final version, 10 pages, to appear in the Journal of Algebraic Combinatorics

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