Context. With the growth of comparative exoplanetology, it is increasingly clear that the relationship between inner and outer planets plays a key role in unveiling the mechanisms governing formation and evolution models. For this reason, it is important to probe the inner region of systems hosting long-period giants in search of undetected lower mass planetary companions.
Aims: We aim to present the results of a high-cadence and high-precision radial velocity (RV) monitoring of three late-type dwarf stars hosting long-period giants with well-measured orbits in order to search for short-period sub-Neptunes (SN, M sin i < 30 M⊕).
Methods: Building on the results and expertise of our previous studies, we carried out combined fits of our HARPS-N data with literature RVs. We used Markov chain Monte Carlo (MCMC) analyses to refine the literature orbital solutions and search for additional inner planets, applying Gaussian process regression techniques to deal with the stellar activity signals where required. We then used the results of our survey to estimate the frequency of sub-Neptunes in systems hosting cold Jupiters, f(SN|CJ), and compared it with the frequency around field M dwarfs, f(SN).
Results: We identify a new short-period, low-mass planet orbiting GJ 328, GJ 328 c, with Pc = 241.8-1.7+1.3 days and Mc sin i = 21.4-3.2+3.4M⊕. We moreover identify and model the chromospheric activity signals and rotation periods of GJ 649 and GJ 849, around which no additional planet is found. Then, taking into account also planetary system around the previously analysed low-mass star BD-11 4672, we derive an estimate of the frequencies of inner planets in such systems. In particular, f(SN|CJ) = 0.25-0.07+0.58 for mini-Neptunes (10 M⊕ < M sin i < 30 M⊕, P < 150 d), marginally larger than f(SN). For lower mass planets (M sin i < 10 M⊕) instead f(SN|CJ) < 0.69, which is compatible with f(SN).
Conclusions: In light of the newly detected mini-Neptune, we find tentative evidence of a positive correlation between the presence of long-period giant planets and that of inner low-mass planets, f(SN|CJ) > f(SN). This might indicate that cold Jupiters have an opposite influence in the formation of inner sub-Neptunes around late-type dwarfs as opposed to their solar-type counterparts, boosting the formation of mini-Neptunes instead of impeding it.

The complete time series of all spectroscopic data used in this work is available at the CDS via anonymous ftp to cdsarc.cds.unistra.fr (ftp://130.79.128.5) or via https://cdsarc.cds.unistra.fr/viz-bin/cat/J/A+A/677/A122

Based on observations made with the Italian Telescopio Nazionale Galileo (TNG), operated on the island of La Palma by the INAF - Fundación Galileo Galilei at the Roque de Los Muchachos Observatory of the Instituto de Astrofísica de Canarias (IAC).