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Low-mass stars in the He-core-burning phase (HeCB) play a major role instellar, galactic, and extragalactic astrophysics. The ability to predictaccurately the properties of these stars, however, depends on our understandingof convection, which remains one of the key open questions in stellarmodelling. We argue that the combination of the luminosity of the AGB bump(AGBb) and the period spacing of gravity modes (DP) during the HeCB phase,provides us with a decisive test to discriminate between competing models ofthese stars. We use the MESA, BaSTI, and PARSEC stellar evolution codes tomodel a typical giant star observed by Kepler. We explore how variousnear-core-mixing scenarios affect the predictions of the above-mentionedconstraints, and we find that DP depends strongly on the prescription adopted.Moreover we show that the detailed behaviour of DP shows the signature of sharpvariations in the Brunt-Vaisala frequency, which could potentially giveadditional information about near-core features. We find evidence for the AGBbamong Kepler targets, and a first comparison with observations shows that, evenif standard models are able to reproduce the luminosity distribution, nostandard model can account for satisfactorily the period spacing of HeCB stars.Our analysis allows us to outline a candidate model to describe simultaneouslythe two observed distributions: a model with a moderate overshooting regioncharacterized by an adiabatic thermal stratification. This prescription will betested in the future on cluster stars, to limit possible observational biases.

Uncertainties on near-core mixing in red-clump stars: effects on the period spacing and on the luminosity of the AGB bump