U torsion structure in Heegaard Floer homology

We construct infinite families of Brieskorn spheres with three singular fibers for which the difference between the largest and the second largest length of U-torsion in their Heegaard Floer homology $H{F}^{-}$ increases to infinity as the family parameter increases. In particular, these families give a negative answer to the strong geography restriction on rational homology spheres. As an application, these families give the examples of integer homology spheres that are neither surgery on a knot nor large surgery on links.