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 $f: V \rightarrow W$ from their restrictions to affine lines, where V, W are ${\mathbb {F}}$-vector spaces and $\dim V \geqslant 2$. First, if $\dim V < |{\mathbb {F}}|$ and $f: V \rightarrow {\mathbb {F}}$ 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 (1850 s) extends beyond bijections: If $f: V \rightarrow W$ preserves affine lines $\ell$, and if $f(v) \not \in f(\ell )$ whenever $v \not \in \ell$, 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], Erdös–Turán [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 $n \geqslant 3$, then one requires affine linearity on at least $\left( {\begin{array}{c}n\\ \lceil n/2 \rceil \end{array}}\right)$-many generic lines to deduce the global affine linearity of f on $R^n$. Moreover, this bound is sharp.