Siegel's Lemma with additional conditions
by
Lenny Fukshansky
University of Texas at Austin

Let K be a number field and N a positive integer. Let W be a non-zero subspace of K^N. Results on existence of a non-zero point of relatively small height (with an explicit bound on height in terms of height of W) in W are usually referred to under the common name of Siegel's Lemma. Such theorems play an important role in Diophantine Approximations. One of the most general and best known theorems in this direction is the one due to Bombieri and Vaaler (1983, Invent. Math.). We generalize this problem in the following way. Let V₁, ... , V_M be proper non-zero subspaces of W. Then we prove the existence of a point of small height in W outside of the union of V₁, ..., V_M, where the bound on height of this point is provided explicitly and depends on heights of W, and V₁, ..., V_M. The bound is essentially best possible (except for possibly the constant). We also obtain a new proof of a version of Siegel's Lemma as a corollary of our method. Another interesting corollary of our main result can be viewed as a converse of Siegel's Lemma. If time permits we will also discuss some generalizations of this result to points of small height outside of a collection of subspaces in non-linear varieties.

Date received: April 3, 2004