the confined directions are practically frozen, and the system should be quasi-low-dimensional, does not work. Counter-intuitively, we find that for $\xi > 0.3$ the LHY correction is practically equal to the one obtained in the 3D case.

Our results provide the working parameters for the planned experiments, which aim at exploring low-dimensional formation of droplets in Bose-Bose mixtures. The quasi-2D regime, as compared to quasi-1D, is accessible for a broader range of $\xi$, i.e., for $\xi \lesssim 0.03$, which is, however, still experimentally demanding. The quasi-1D regime is attained for a much smaller range of $\xi$, i.e., $\xi\lesssim0.0004$, which poses a severe experimental constraint. However, our work reveals that yet unexplored 3D-1D crossover supports exotic droplets, different from both 3D and quasi-1D case, and formed only due to quantum fluctuations. Such droplets also exist at the border of the quasi-2D regime. The results we present pave the way for exploring new states of matter in low-dimensional systems, in which quantum fluctuations play the prominent role.

P.Z. and Z.I. acknowledge the support from the Polish National Science Center Grant No. 2015/17/B/ST2/00592. M.P. and T.W. were supported by the Polish National Science Center Grant No. 2014/14/M/ST2/00015. M.G. acknowledges support from the (Polish) 