Inhomogeneous Problems

In 1901, Minkowski proved that if \(L_{1}(x_1, x_2) = \alpha x_1 + \beta x_2, L_{2}(x_1, x_2) = \gamma x_1 + \delta x_2\) are two real linear forms of determinant \(\Delta = \alpha \delta - \beta \gamma \ne 0,\) then given any real numbers \(c_1, c_2 \) there exist integers \(u_1 ,u_2 \) satisfying \(| L_{1} (u_1, u_2) + c_1 | | L_{2} (u_1, u_2) + c_2 | \le \frac{1}{4} |\Delta |.\) Minkowski is believed to have conjectured that its generalization to \(n\le 10\) with the efforts of several mathematicians. Different approaches to tackle Minkowski’s conjecture and its history are discussed in this chapter. Several different proofs of Minkowski’s result for \(n=2\) had been obtained by various mathematicians in an effort to find a method which would generalize to higher dimensions. Here we give a proof of Minkowski’s Theorem for \(n=2\) given by Davenport in 1948, and a proof of Minkowski’s Conjecture for \(n=3\) given by Remak and Davenport in 1939. The approach of Remak-Davenport consists of two parts called Conjecture I and Conjecture II. Conjecture I has been proved by C. T. McMullen in 2005 for all \(n\ge 3\) . Conjecture II was proved by Woods for \(n=4,5\) and 6 in 1965 and 1972. Following Woods’ method Conjecture II has been proved for \(n=7, 8\) by Hans-Gill, Raka and Sehmi (2009, 2011). For \(n= 9\) and 10 it is settled by Kathuria and Raka (2016, 2022). In 2017, Regev et. al showed that Conjecture II is false for \(n\ge 30\) and then in 2019, Chen and Xu proved its falsehood for \(n\ge 24\) . In this chapter we shall give a proof of Conjecture II for \(n=2,3\) and 4 following Woods’ method. We shall also give a proof of a weaker result by \(\mathrm{\check{C}}\) ebotarev (1940) on the product of non-homogeneous linear forms and discuss some other related results including Chalk’s Theorem. Minkowski’s Theorem for \(n=2\) can be reformulated in terms of non-homogeneous indefinite binary quadratic forms. This can be generalized to inhomogeneous minima of indefinite quadratic forms in n variables. In the last section of this chapter, we shall discuss Watson’s Conjecture (1962) on non-homogeneous minima of indefinite quadratic forms in n variables and a conjecture of Bambah, Dumir and Hans-Gill (1984) on positive values of non-homogeneous indefinite quadratic forms.