Recovering affine linearity of functions from their restrictions to affine lines

Motivated by recent results of Tao–Ziegler [Discrete Anal. 2016] and Greenfeld–Tao (2022 preprint) on concatenating affine-linear functions along subgroups an abelian group, we show three recovering affine linearity $$f: V \rightarrow W$$ from their restrictions to lines, where V, W are $${\mathbb {F}}$$ -vector spaces $$\dim \geqslant 2$$ . First, if < |{\mathbb {F}}|$$ {\mathbb is when restricted lines parallel a basis certain “generic” through 0, then f V. (This extends all modules M over unital commutative rings R with large enough characteristic.) Second, explain how classical result attributed von Staudt (1850 s) beyond bijections: If preserves $$\ell $$ , $$f(v) \not \in f(\ell )$$ whenever $$v \ell this also suffices recover but up field automorphism. In particular, prime {Z}}/p{\mathbb {Z}}$$ ( $$p>2$$ ) or {Q}}$$ completion {Q}}_p$$ {R}}$$ We quantitatively refine our first above, via weak multiplicative variant the additive $$B_h$$ -sets initially explored Singer [Trans. Amer. Math. Soc. 1938], Erdös–Turán [J. London 1941], Bose–Chowla [Comment. Helv. 1962]. Weak occur inside characteristic, in infinite finite integral domains/fields. that among any these classes rings, $$M = R^n$$ for some $$n 3$$ one requires at least $$\left( {\begin{array}{c}n\\ \lceil n/2 \rceil \end{array}}\right) -many generic deduce global $$R^n$$ Moreover, bound sharp.