The number of configurations in lattice point counting. II (Q2872029)

Let \(S\) be a fixed closed bounded set in the plane and consider the shifted dilations \(S(t,u,v)=tS+(u,v)\) where \(t>0\), \(u\) and \(v\) are real. Let \(J_S(t,u,v)\) be the set of integer lattice points in \(S(t,u,v)\). As \(t,u\) and \(v\) vary one gets different sets \(J_S(t,u,v)\), and these are said to be equivalent if they differ by translation by an integral vector. Thus if \(S\) is the closed unit square, and \(t\) is fixed, then \(J_S(t,u,v)\) may belong to one of 4 equivalence classes. Specifically, if \(r=[t]\), one can get \(r\times r\) or \((r+1) \times(r+1)\) square arrays of integer lattice points, or \(r\times(r+1)\) or \((r+1)\times r\) rectangular arrays.NEWLINENEWLINEThe paper is concerned with the number \(L_S(N)\) of equivalence classes for \(J_S(t,u,v)\) (as \(t, u\) and \(v\) vary) containing exactly \(N\) points, and the number \(M_S(N)\) with at most \(N\) points. For example, when \(S\) is the unit square one sees that \(L_S(N)\) takes the values \(0,1\) or \(2\).NEWLINENEWLINEIt is shown that \(L_S(N)\leq 2N-1\) and \(M_S(N)\leq N^2\) whenever \(S\) is convex, and that equality holds in both cases if \(S(t,u,v)\) never contains 4 or more integer vectors on its boundary. This latter condition fails if \(S\) is a disc. In this case it is conjectured that \(L_S(N)\sim 2N\), but the numerical evidence presented, for \(N\leq 155\), is unconvincing.NEWLINENEWLINEA second theorem shows (in a precise sense) that it is rare for \(S(t,u,v)\) to contain 4 or more integer vectors on its boundary.NEWLINENEWLINEFor Part I, see Forum Math. 22, No. 1, 127--152 (2010; Zbl 1206.11124).