Fix a positive integer $k>0$. For $p>k$ a prime, let $A\_p$ be a subset of the finite field $\mathbb{Z}/p\mathbb{Z}$ of size $k$ which contains a primitive element.


Define $G\_p$ to be the (di)graph whose vertices are elements of $\mathbb{Z}/p\mathbb{Z}$, with two vertices $i,j$ joined by an edge provided $j=ia$ or $j=i+a$ for some $a\in A\_p$. 


(I'm mainly interested in the situation where $A\_p$ is closed under the operations of taking multiplicative and additive inverses; under these assumptions I can think of $G\_p$ as a graph rather than a digraph.)


**Question: Is $(G\_p)\_{p \textrm{ a prime}}$ a family of expanders?**




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Background: I'm expecting the answer to be either "possibly" or "no" (because if it were "yes" I'd hope I'd have heard about it already).


My interest comes in studying the Bourgain-Gamburd machinery for proving expansion from results about growth. For the family $(G\_p)$, the relevant growth result is the Bourgain-Katz-Tao sum-product theorem for fields of prime order.


One needs more than just a growth result of course, one also needs to have some notion of `quasirandomness' (but I think I can handle this), as well as a lower bound on the girth of the graph. I've not thought much about this last aspect so I guess this is the most likely to be the sticking point.



Constructing expanders in Z/pZ
No, because solvable groups are amenable. You're asking: Is is there a set in Z/pZ almost invariant by x->x+1 and 2x? Here's one: take the union of I, I/2, .., I/2^n, where I is an interval of length much bigger than 2^n. 


