A note on distinct distances in rectangular lattices

In his famous 1946 paper, Erdős (1946) proved that the points of a n×n portion of the integer lattice determine Θ(n/logn) distinct distances, and a variant of his technique derives the same bound for n×n portions of several other types of lattices (e.g., see Sheffer (2014)). In this note we consider distinct distances in rectangular lattices of the form {(i,j)∈Z2∣0≤i≤n1−α,0≤j≤nα}, for some 0<α<1/2, and show that the number of distinct distances in such a lattice is Θ(n). In a sense, our proof “bypasses” a deep conjecture in number theory, posed by Cilleruelo and Granville (2007). A positive resolution of this conjecture would also have implied our bound.