Improved $L^p$ bounds for Bochner--Riesz operators associated with rough convex domains in the plane

We consider generalized Bochner-Riesz operators associated with planar convex domains. We construct domains which yield improved $L^p$ bounds for the Bochner-Riesz over previous results of Seeger-Ziesler and Cladek. The constructions are based on the existence of large $B_m$ sets due to Bose-Chowla, and large $\mathrm{\Lambda }(p)$ sets due to Bourgain.