On Helly numbers of exponential lattices

Given a set S⊆R², define the Helly number of S, denoted by H(S), as the smallest positive integer N, if it exists, for which the following statement is true: for any finite family F of convex sets in R² such that the intersection of any N or fewer members of F contains at least one point of S, there is a point of S common to all members of F. We prove that the Helly numbers of exponential lattices {αⁿ:n∈N₀}² are finite for every α>1 and we determine their exact values in some instances. In particular, we obtain H({2ⁿ:n∈N₀}²)=5, solving a problem posed by Dillon (2021). For real numbers α,β>1, we also fully characterize exponential lattices L(α,β)={αⁿ:n∈N₀}×{βⁿ:n∈N₀} with finite Helly numbers by showing that H(L(α,β)) is finite if and only if log_α(β) is rational.