Hyperbolic Lebesgue Constants in Dimension Three

For the three-dimensional hyperbolic crosses {k∈Z3,α,β,γ>0:1≤|k1|α|k2|β|k3|γ≤Rα+β+γ}, three types of estimates (from above) of the Lebesgue constants generated by them are possible depending on α, β, and γ. We improve the most problematic of them by showing that for R→∞, it may be RlnR rather than the previously known RlnR.