Abstract:
: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.
journal_name
Proc Natl Acad Sci U S Aauthors
Chudnovsky GVdoi
10.1073/pnas.80.10.3138subject
Has Abstractpub_date
1983-05-01 00:00:00pages
3138-41issue
10eissn
0027-8424issn
1091-6490journal_volume
80pub_type
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