A Lozenge triangulation of the plane with integers (Q6632414)

Consider 3 transformations \(H',H'',H'''\):\(\mathbb{Z}^3\to\mathbb{Z}^3\) that each leaves two components unchanged while the negative third adds 1 and the other two. Denote by \(H^{[n]}\) a set of transformation obtained by \(n\) compositions of \(H',H'',H'''\) in all possible orders. For any \((a,b,c)\in\mathbb{Z}^3\) denote by \(\mathcal{R}_H(a,b,c)\) the set of all natural \(x\) such that \(x\) is an element of triple from the set of triples \(H^{[n]}(a,b,c)\) for some \(n\).\N\NThe main results are following.\N\N1. For any \((a,b,c)\in\mathbb{Z}^3\) there exists \(h\in\mathbb{Z}\) such that either \(\mathcal{R}_H(a,b,c)=\mathcal{R}_H(0,0,0)+h\) or \(\mathcal{R}_H(a,b,c)=\mathcal{R}_H(0,1,1)+h\).\N\N2. For any fixed \(M\in\mathbb{Z}\) \(\mathcal{R}_H(a,b,c) \cap (-\infty,M]\) is finite for any \((a,b,c)\in\mathbb{Z}^3\).\N\N3. For any prime \(p\) the limit densities of residue classes \(u=R\pmod p\) of the represented integers in \(R\in \mathcal{R}_H(a,b,c)\) are calculated.\N\NThe main tools of proofs are geometric representation of \(\mathcal{R}_H(a,b,c)\) by lozenge triangulation of the plane with integers on vertices and characterization of the represented integers using families of certain quadratic forms.