Let $\mathbb{F}_q$ be a finite field of $q$ elements, let $\chi$ be a non-trivial additive character of $\mathbb{F}_q$, and let $\psi$ be a non-trivial multiplicative character of $\mathbb{F}_q$. Also, let $a(x)$ be a linear recurrence of order $n$ with period $t$ in $\mathbb{F}_q$.
It is known [1, Chapter 5] that the upper bounds $$\left|\sum_{x=1}^t \chi(a(x))\right| \leq q^{n/2}$$ and $$\left|\sum_{x=1}^t \psi(a(x))\right| \leq q^{(n-1)/2} |G_\mu|$$ hold (here $G_\mu$ is the group of multipliers of $a(x)$, that is, the group of $v \in \mathbb{F}_q^*$ such that there exists an integer $h_v \geq 0$ for which $a(x + h_v) = v\, a(x)$ for every integer $x \geq 1$).
If $b(x)$ is another linear recurrence of order $n$ with period $t$ in $\mathbb{F}_q$, what is known about upper bounds for $$\left|\sum_{x=1;b(x) \neq 0}^t \chi(a(x)/b(x))\right|$$ and $$\left|\sum_{x=1;b(x) \neq 0}^t \psi(a(x) / b(x))\right| \quad ?$$
For the second quantity, one can write
$$\left|\sum_{x=1\;b(x) \neq 0}^t \psi(a(x) / b(x))\right| = \left|\frac1{t}\sum_{j=1}^t\sum_{x=1}^t \psi(a(x))e(jx/t) \overline{\sum_{y=1}^t \psi(b(y))e(jy/t)}\right|$$ $$\leq \max_{j} \left|\sum_{x=1}^t \psi(a(x))e(jx/t)\right| \left|\sum_{y=1}^t \psi(b(y))e(jy/t)\right| \leq q^{n-1}|G_\mu|^2 ,$$ using the bound $|\left|\sum_{x=1}^t \psi(a(x))e(jx/t)\right| \leq q^{(n-1)/2}|G_\mu|$ of [2]. However, this is quite weak. I guess it can be improved.
Thanks for any reference/suggestion.
[1] Everest, van der Poorten, Shparlinski, and Ward, Recurrence Sequences, 2003
[2] Shparlinski, Distribution of nonresidues and primitive roots in recurrent sequences, Mathematical notes of the Academy of Sciences of the USSR volume 24, pages 823–828 (1978)
