On the structure of the hh-fold sumsets

Let~$A$ be a set of nonnegative integers. Let~$(h A)^{(t)}$ be the set of all integers in the sumset~$hA$ that have at least~$t$ representations as a sum of~$h$ elements of~$A$. In this paper, we prove that, if~$k \geq 2$, and~$A=\left\{a_{0}, a_{1}, \ldots, a_{k}\right\}$ is a finite set of integers such that~$0=a_{0}<a_{1}<\cdots<a_{k}$ and $\gcd\left(a_{1}, a_2,\ldots, a_{k}\right)=1,$ then there exist integers ~$c_{t},d_{t}$ and sets~$C_{t}\subseteq[0, c_{t}-2]$, $D_{t} \subseteq[0, d_{t}-2]$ such that $$(h A)^{(t)}=C_{t} \cup\left[c_{t}, h a_{k}-d_{t}\right] \cup\left(h a_{k-1}-D_{t}\right)$$ for all~$h \geq\sum_{i=2}^{k}(ta_{i}-1)-1.$ This improves a recent result of Nathanson with the bound $h \geq (k-1)\left(t a_{k}-1\right) a_{k}+1$.Comment: 8 page

Vortex-lattice structures in rotating Bose-Fermi superfluid mixtures

The system of Bose-Fermi superfluid mixture offers a playground to explore rich macroscopic quantum phenomena. In a recent experiment of Yao {\it et al.} [Phys. Rev. Lett. {\bf 117}, 145301 (2016)], $^{41}$K-$^{6}$Li superfluid mixture is implemented. Coupled quantized vortices are generated via rotating the superfluid mixture, and a few unconventional behaviors on the formations of vortex numbers are observed, which can be traced to boson-fermion interactions. Here we provide a theoretical insight into the unconventional behaviors observed in the experiment. To this end, the orbital-free density functional theory is hired, and its utility is validated by making comparison of numerical results and full microscopic theory for vortex lattices in strongly interacting Fermi superfluids alone. We also predict interesting phenomena which can be readily explored experimentally, including the novel structures of vortex lattices in Bose-Fermi superfluid mixtures in phase-separated regimes, and attractive interactions between vortex lines belonging to distinct superfluids.Comment: 27 pages, 9 figures, Submission to SciPost Physic