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Let $K$ be a finite field with $p$ elements.

(a) Let $f\in K\lbrack x\rbrack$ be such that (i) $\deg(f)<p$ and (ii) $f(2x) = f(x)$ for $\geq (1-\epsilon) p$ values of $x$ in $K$. What can we say about $f$?

(b) Let $F\in K\lbrack x,y\rbrack$ be such that (ii) $\deg(F)\leq \sqrt{2 p}$ and (ii) the curves $F(x,y)=0$, $F(2x,y)=0$ and $F(4x,y)=0$ intersect at $\geq (1-\epsilon) p$ points $(x,y)$ (in $\mathbb{A}^2(K)$ or in $\mathbb{A}^2(\overline{K})$ -- it's equivalent). What can you conclude about $F$?

Here is a partial answer to (a) that does not quite satisfy me. Write $f=\sum_{j=0}^{p-1} a_j x^j$. Then the function $f(2x)-f(x) = \sum_{j=0}^{p-1} (2^j-1) a_j x^j$ takes the value $0$ at all but $\epsilon p$ points of K. Hence, $f(2x)-f(x)$ is of the form $P(x) (x^p-x)/\prod_{s\in S} (x-s)$, where $\deg(P)\leq \epsilon p$ and $S$ is a subset of $K$ with $\leq \epsilon p$ elements. (Of course, this says that $f(2x)-f(x)$ has a nice form, but doesn't quite give us a nice form for $f(x)$, at least not that I can see it.)

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