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.