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$ \newcommand{\F}{{\mathbb F}}$ $ \newcommand{\R}{{\mathbb R}}$ $ \renewcommand{\phi}{\varphi}$ Let $ p\ge 5$ be a prime. As an easy exercise, if the functions $ \phi_1,\phi_2,\phi_3\colon\F_p\to\R$ satisfy $ \phi_1(x)+\phi_2(y)=\phi_3(x+y)$ for all pairs $ (x,y)\in\F_p^2$ , then they are constant functions. Given that $ \phi_1,\phi_2,\phi_3\colon\F_p\to\R$ are non-constant, what is the largest possible number of pairs $ (x,y)\in\F_p^2$ satisfyingRead more

Let $ p$ be a (large) prime. How large can a set $ C\subset\mathbb F_p$ be given that there is a function $ f\colon\mathbb F_p^\times\to\mathbb F_p$ such that for every element $ g\in \mathbb F_p$ , there is at most one element $ c\in C$ with the property that $ $ g \notin \{f(z)-cz\colon z\in\mathbbRead more