Estimates for Character Sums with Various Convolutions

Abstract: We provide estimates for sums of the form [\left|\sum_{a\in A}\sum_{b\in B}\sum_{c\in C}\chi(a+b+c)\right|] and [\left|\sum_{a\in A}\sum_{b\in B}\sum_{c\in C}\sum_{d\in D}\chi(a+b+cd)\right|] when $A,B,C,D\subset \mathbb F_p$, the field with $p$ elements and $\chi$ is a non-trivial multiplicative character modulo $p$.