Estimating inner product over $[r]^d$ Alice has a vector $x \in [r]^d$ and Bob has $y \in [r]^d$, where $[r] \stackrel{\rm def}{=} \{0,1,\dots,r\}$. Alice send a message $M(x)$ to Bob and Bob wants to estimate the inner product $\left&lt;x,y\right&gt;$ with multiplicative error $\epsilon$ with probability $1 - \delta$.

My question is. Is there any known lower bound on $|M(x)|$? My intuition tells me that Alice pretty much has to estimate each coordinate of $x$ up to $\epsilon$ precision since Bob can have an input that zeros out all but one of the entry. Does that give you something like $\Omega(d \log 1/\epsilon)$?
In the indexing problem Alice has a vector $x \in \{0,1\}^d$ and Bob has a number $i$, and Bob wants to learn $x_i$. The randomized one-way communication complexity of this problem is $\Omega(d)$ (see Section 3 of this paper), and it can be solved by approximating $x_i = \langle x, e_i\rangle$ up to any multiplicative factor where $e_i$ is the vector with $1$ in the $i$-th coordinate and $0$'s everywhere else. So your problem has $\Theta(d)$ randomized one-way communication complexity for $r=1$.