Chebotarev density theorem in short intervals for extensions of ${\mathbb{F}}_q(T)$

An old open problem in number theory is whether Chebotarev density theorem holds in short intervals. More precisely, given a Galois extension $E$ of $\mathbb{Q}$ with Galois group $G$, a conjugacy class $C$ in $G$ and an $1\geq \varepsilon>0$,onewantstocomputetheasymptoticofthenumberofprimes$x\leq p\leq x+x^{\varepsilon}$withFrobeniusconjugacyclassin$E$equalto$C$\mathrm{.}Thelevelofdifficultygrowsas$\varepsilon$ becomes smaller. Assuming the Generalized Riemann Hypothesis, one can merely reach the regime $1\ge \epsilon >1\mathrm{/}2$. We establish a function field analogue of Chebotarev theorem in short intervals for any $\epsilon >0$. Our result is valid in the limit when the size of the finite field tends to $\mathrm{\infty }$ and when the extension is tamely ramified at infinity. The methods are based on a higher dimensional explicit Chebotarev theorem, and applied in a much more general setting of arithmetic functions, which we name $G$-factorization arithmetic functions.