Cancellation in a short exponential sum

Let q be an odd integer, let τ be the order of 2 modulo q and let ξ be a primitive qth root of unity. Upper bounds for are proved in terms of the parameters μ and ν when q diverges along sequences Sμ,ν for which the quotient τ/log2q belongs to the interval [μ,ν], with 1⩽μ and ν close enough to 1.