Let $p$ be a (large) prime.



> 
> Does there exist a function $f\colon\mathbb F\_p^\times\to\mathbb F\_p$ such that the three sets
>  $$ \{f(z)-z\colon z\in\mathbb F\_p^\times\},\ \{f(z)\colon z\in\mathbb F\_p^\times\},\ \text{and}\ \{f(z)+z\colon z\in\mathbb F\_p^\times \} $$
>  form a double-cover of $\mathbb F\_p$ (in the sense that any element of $\mathbb F\_p$ lies in at least two of them)?
> 
> 
> 


More generally, 



> 
> Do there exist a function $f\colon\mathbb F\_p^\times\to\mathbb F\_p$ and elements $c\_1,c\_2,c\_3\in\mathbb F\_p$ such that the images of the functions $z\mapsto f(z)-c\_iz\ (z\in\mathbb F\_p^\times)$ double-cover $\mathbb F\_p$?
> 
> 
> 


I hardly believe this is a tough problem; rather, I suspect I am overlooking a simple argument.



Double-covering $\mathbb F_p$ with three sets
Start with $f(z)=3z$, then all non-zero elements are covered three times. Change $f(a)$ to 0 and $f(b)$ to $b$. Now 0 is covered two times and $2a,3a,4a$, $2b,3b,4b$ are covered two times also (assume that all these 6 numbers are different, this is possible for large $p$). Some other multiplicities could increase (and they did), but this is ok for us.


