Estimating inner product over $[r]^d$
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 $r=1$.