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We say that a function $ f:{\mathbb Z}\times {\mathbb Z} \to {\mathbb Z}$ has bounded neighbor-difference if there is an integer $ K\in\mathbb{N}$ such that for all $ x,y\in{\mathbb Z}$ we have $ $ |f(x,y) – f(x+1,y)| \leq K \text{ and } |f(x,y) – f(x,y+1)| \leq K.$ $ Is it possible that a function $Read more

$ \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

$ \newcommand{\F}{{\mathbb F}}$ Let $ q$ be an odd prime power. A blocking set in the affine plane $ \F_q^2$ is a set blocking (meeting) every line. A union of two non-parallel lines is a blocking set of size $ 2q-1$ , and it is well-known that this is the smallest possible size of aRead more

The Golomb space $ \mathbb G$ is the set $ \mathbb N$ of positive integers endowed with the topology generated by the base consisting of arithmetic sequences $ a+b\mathbb N_0:=\{a+bn:n\ge 0\}$ with $ a,b$ relatively prime. It is known that the Golomb space does not admit non-trivial homeomorphisms. On the other hand, $ \mathbb G$Read 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