In the setting of regularly varying time series, a cluster of exceedances is a short period for which the supremum norm exceeds a high threshold. We propose to study a generalization of this notion considering short periods, or blocks, with lp−norm above a high threshold. We derive large deviation principles of blocks and apply these results to improve cluster inference. We focus on blocks estimators and show they are consistent when we use large empirical quantiles from the lp −norm of blocks as threshold levels. We derive an adaptive threshold selection method for cluster inference in lp. Our approach focuses on the case p < ∞ rather than the classical one for p = ∞ where the bias is more difficult to control.