Rational approximations to linear forms of exponentials and binomials.
Mahler proved the following quantitative result supplementing the Lindemann-Weierstrass theorem: Sigma(i=0) (n)C(i)e(ri) > H(-n-epsilon) for any distinct rational numbers r(0),r(1),..., r(n) and rational integers C(0),C(1),...,C(n) with H = max(0</=i</=n) C(i). We improve Mahler's estimate by replacing exponentials e(ri) by linearly independent linear forms L(i) = Sigma L(ij)e(sij) with rational L(ij),s(ij)i = 0,1,...,n. Similar results are obtained for binomials (a/b)(ri) or Sigma L(ij)(a/b)(sij) with integers a,b and logb/loga > 1 - epsilon. The simplest examples of new numbers with the irrationality exponent "2 + epsilon" are sinh 1 or sin 1.
Department of Mathematics, Columbia University, New York, New York 10027.
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Chudnovsky