 Cyclotomic class of a finite field

The cosets of the index-$m$ subgroup of the multiplicative group of a finite field are its cyclotomic classes of index $m$. For a character with $\chi (\gamma )={\zeta }_m$, their indicators are $1_{C_j}(x)=m^{-1}{\sum }_{h=0}^{m-1}{\zeta }_m^{-jh}{\chi }^h(x)$, with every character extended by zero at zero. Jacobi sums of finite-field characters imply $|C_i\cap (t+C_j)|=q/m^2+O_m(\sqrt{q})$ uniformly for $t\ne 0$.