Sums of $\|\theta\|^{-1}$ for $\delta$-separated points

Question

In the setting of the classical circle method we know bounds for quantities of the form $$\sum_i \min (A, \|\theta_i\|^{-1})$$ typically when the $\theta_i$ are in an arithmetic progression (and $\|\cdot\|$ denotes the distance to the closest integer). Can we have finer bounds in the case the $\theta_i$ are $\delta$-separated (modulo 1)? Typically I am puzzled by a bound of the form $$\sum_{i<d} \min(A, \|i(k/d + \beta)\|^{-1}) \ll d \log d$$ plus the sum of the summands. I do not have any proof (or feeling) for this result.