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 / log n ) 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 ) ∈ Z 2 ∣ 0 ≤ i ≤ n 1 − α , 0 ≤ j ≤ n α } , 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.