An inverse and a stability result for Ruzsa's inequality on triple sumsets

Ruzsa’s inequality states that $\mathrm{\mid }A+A+A\mathrm{\mid }\le \mathrm{\mid }A+A{\mathrm{\mid }}^{3\mathrm{/}2}$ for any finite set $A$ in a commutative group. Ruzsa has constructed examples showing that this inequality is sharp asymptotically, up to a constant factor. We prove an inverse result which says that if $\mathrm{\mid }A+A+A\mathrm{\mid }\ge \frac{1}{M}\mathrm{\mid }A+A{\mathrm{\mid }}^{3\mathrm{/}2}$ for some parameter $M,$ then the set $A$ resembles the sets in Ruzsa’s construction. We then construct more families of examples which suggest that our inverse result is likely best possible qualitatively. The method extends to give an inverse result for a higher sumset analogue of Ruzsa’s inequality, namely $\mathrm{\mid }(h+1)A\mathrm{\mid }\le \mathrm{\mid }hA{\mathrm{\mid }}^{\frac{h+1}{h}}$ for any $h\ge 2.$ We also provide a “99%-stability” version of Ruzsa’s inequality, which describes near optimal structures when $M$ is very close to $1.$