ABSTRACT
Suppose that A is an M × N real matrix. We view A as the matrix of coefficients of M linear forms in N variables. In the theory of Diophantine approximation it is well-known that:
If there are no “small” integer vectors x → https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203747018/bef25d74-ef02-4914-ac4f-27ec89fc30ac/content/eq268.tif"/> such that A x → https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203747018/bef25d74-ef02-4914-ac4f-27ec89fc30ac/content/eq269.tif"/> is “near” an integer vector, then for any β → ∈ ℝ M https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203747018/bef25d74-ef02-4914-ac4f-27ec89fc30ac/content/eq270.tif"/> there must exist a “small” integer vector x → https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203747018/bef25d74-ef02-4914-ac4f-27ec89fc30ac/content/eq271.tif"/> such that A x → − β → https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203747018/bef25d74-ef02-4914-ac4f-27ec89fc30ac/content/eq272.tif"/> is “near” an integer vector.
