Induced arithmetic removal: complexity 1 patterns over finite fields

We prove an arithmetic analog of the induced graph removal lemma for complexity 1 patterns over finite fields. Informally speaking, we show that given a fixed collection of $r$-colored complexity 1 arithmetic patterns over ${\mathbb{F}}_q$, every coloring $\phi \colon \mathbb F_q^n \setminus{0} \to [r]$with$o(1)$densityofeverysuchpatterncanberecoloredonan$o(1)$-fraction of the space so that no such pattern remains.