# Approximate homomorphisms

Let $f:Z_2^n \to Z_p$ be a one-to-one map, where say $2^n<p<2^{n+1}$. What is the maximal probability that $\Pr[f(x+y)=f(x)+f(y)]$ where $x,y \in Z_2^n$ are uniform and independent? The identity map (treating bits as base-two representation) gives probability of $(3/4)^n$. Is it true that this probability for any $f$ is always $\le 2^{-cn}$ for some absolute $c>0$?

Also, the same question, but in the other direction: $f:Z_p \to Z_2^n$, where here $2^{n-1}<p<2^n$. Is the probability always $\le p^{-c}$ for some $c>0$?

All the references I found show that these probabilities are somewhat smaller than 1, but I suspect that they should be polynomially small in the group size. This might be well known - I appreciate any reference.

• The techniques in this paper arxiv.org/pdf/1308.2247v1.pdf might be useful. Jun 4, 2015 at 0:50
• What exactly? the paper seems to study problems in characteristics zero Jun 4, 2015 at 4:06
• I was thinking of the ideas around Corollary 2.4 in that paper. (And also that ${\Bbb Z}/p$ for large enough $p$ might behave not too differently from ${\Bbb Z}$.) Anyway, just a quick thought. Jun 4, 2015 at 4:21

Lemma: Suppose that $p$ is an odd prime and $f: \mathbb{Z}/p\mathbb{Z}\rightarrow \mathbb{F}_2^n$ is injective. Then \begin{equation*} \mathbb{P}(f(x+y)=f(x)+f(y))=p^{-\Omega(1)}. \end{equation*}
Proof: Write $\Gamma:=\{(x,f(x)):x \in \mathbb{Z}/p\mathbb{Z}\} \subset \mathbb{Z}/p\mathbb{Z} \times \mathbb{F}_2^n$ and $\varepsilon$ for the probability we are interested in. Using the language of summation instead of expectation the says \begin{equation*} \langle 1_\Gamma\ast 1_\Gamma,1_\Gamma\rangle_{\ell_2} = \varepsilon p^2, \end{equation*} where \begin{equation*} g\ast h(x):=\sum_{y+z=x}{g(y)h(z)} \text{ and }\langle g,h\rangle_{\ell_2}:=\sum_x{g(x)\overline{h(x)}}, \end{equation*} for functions $g$ and $h$.
Since $f$ is a function, $|\Gamma|=p$, and so by Cauchy-Schwarz in the usual way we get \begin{equation*} \|1_\Gamma\ast 1_\Gamma\|_{\ell_2}^2 \geq \frac{1}{|\Gamma|}\langle 1_\Gamma\ast 1_\Gamma,1_\Gamma\rangle_{\ell_2}^2 \geq \varepsilon^2 \frac{p^4}{|\Gamma|} = \varepsilon^2 |\Gamma|^3. \end{equation*} The Balog-Szemerédi-Gowers lemma then tells us that there is a set $\Gamma' \subset \Gamma$ with $|\Gamma'| \geq \varepsilon^{O(1)}|\Gamma|$ and \begin{equation*} |\Gamma'+\Gamma'| \leq \varepsilon^{-O(1)}|\Gamma'|. \end{equation*} For a set $\Lambda \subset \mathbb{Z}/p\mathbb{Z} \times \mathbb{F}_2^n$ write $2\cdot \Lambda:=\{(2x,2y): (x,y) \in \Lambda\}$. Since $p \neq 2$ the map \begin{equation*} \Gamma' \rightarrow 2\cdot \Gamma'; (x,f(x)) \mapsto (2x,2f(x)) \end{equation*} is a bijection and hence $|2\cdot \Gamma'| = |\Gamma'|=\varepsilon^{O(1)}p$. Now, by Cauchy-Schwarz again we have $$\tag{1}\label{eq:1} \|1_{\Gamma'} \ast 1_{2\cdot \Gamma'}\|_{\ell_2}^2 \geq \frac{(|\Gamma'||2\cdot \Gamma'|)^2}{|\Gamma'+2\cdot\Gamma'|} = \varepsilon^{O(1)}p^3,$$ where the last inequality follows from the fact that \begin{equation*} |\Gamma'+2\cdot \Gamma'| \leq |\Gamma' + \Gamma' + \Gamma'| \leq \varepsilon^{-O(1)}|\Gamma'| = \varepsilon^{-O(1)}p \end{equation*} by Plünnecke's inequality.
On the other hand, if $(a,f(a)),(b,f(b)) \in \Gamma'$ and $(2c,2f(c)),(2d,2f(d)) \in 2\cdot\Gamma'$ have $$\tag{2}\label{eq:2} (a,f(a))+(2c,2f(c))=(b,f(b)) +(2d,2f(d)),$$ then $a+2c=b+2d$ and $f(a)+2f(c)=f(b)+2f(d)$. But the image of $f$ is in $\mathbb{F}_2^n$ and so $2f(c)=0=2f(d)$ whence $f(a)=f(b)$. On the other hand $f$ is injective so $a=b$. It follows from this (and the equation $a+2c=b+2d$) that $2c=2d$, but $p$ is odd and so $c=d$. Thus there are at most $p^2$ quadruples such that \eqref{eq:2} holds. These quadruples are precisely what is counted by the left hand side of \eqref{eq:1} and so \begin{equation*} \varepsilon^{O(1)}p^3 \leq p^2, \end{equation*} from which we get $\varepsilon = p^{-\Omega(1)}$ as claimed.