Recovering affine linearity of functions from their restrictions to affine lines

Khare, A and Tikaradze, A (2023) Recovering affine linearity of functions from their restrictions to affine lines. In: Journal of Algebraic Combinatorics .

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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→ W from their restrictions to affine lines, where V, W are F -vector spaces and dim V⩾ 2 . First, if dim V< | F| and f: V→ 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→ W preserves affine lines ℓ , and if f(v) ∉ f(ℓ) whenever v∉ ℓ , then this also suffices to recover affine linearity on V, but up to a field automorphism. In particular, if F is a prime field Z/ pZ (p> 2) or Q , or a completion Qp or R , then f is affine-linear on V. We then quantitatively refine our first result above, via a weak multiplicative variant of the additive Bh -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 Bh -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= Rn for some n⩾ 3 , then one requires affine linearity on at least (n⌈n/2⌉) -many generic lines to deduce the global affine linearity of f on Rn . Moreover, this bound is sharp.

Item Type:	Journal Article
Publication:	Journal of Algebraic Combinatorics
Publisher:	Springer
Additional Information:	The copyright for this article belongs to the Springer
Keywords:	Affine-linear maps; Bh set; Concatenation; Field automorphism; Sidon set.
Department/Centre:	Division of Physical & Mathematical Sciences > Mathematics
Date Deposited:	15 Jul 2023 07:11
Last Modified:	15 Jul 2023 07:11
URI:	https://eprints.iisc.ac.in/id/eprint/82480

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