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 Q with Galois group G, a conjugacy class C in G and an 1 ≥ ε > 0, one wants
to compute the asymptotic of the number of primes x ≤ p ≤ x + x
ε with Frobenius conjugacy class in E equal to C. The level of difficulty grows as ε becomes
smaller. Assuming the Generalized Riemann Hypothesis, one can merely reach
the regime 1 ≥ ε > 1/2. We establish a function field analogue of Chebotarev
theorem in short intervals for any ε > 0. Our result is valid in the limit when
the size of the finite field tends to ∞ 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.