Sums and Products of Distinct Sets and Distinct Elements in C
نویسنده
چکیده
Let A and B be finite subsets of C such that |B| = C|A|. We show the following variant of the sum product phenomenon: If |AB| < α|A| and α ! log |A|, then |kA+ lB|" |A|k|B|l. This is an application of a result of Evertse, Schlickewei, and Schmidt on linear equations with variables taking values in multiplicative groups of finite rank, in combination with an earlier theorem of Ruzsa about sumsets in Rd. As an application of the case A = B we give a lower bound on |A+|+ |A×|, where A+ is the set of sums of distinct elements of A and A× is the set of products of distinct elements of A.
منابع مشابه
2 0 Fe b 20 09 Sums and Products of Distinct Sets and Distinct Elements in Fields of Characteristic 0
X iv :0 90 2. 35 06 v1 [ m at h. C O ] 2 0 Fe b 20 09 Sums and Products of Distinct Sets and Distinct Elements in Fields of Characteristic 0 Karsten O. Chipeniuk 1 Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, B.C. V6T 1Z2 Canada Abstract Let A and B be finite subsets of an algebraically closed field K of characteristic 0 such that |B| = C|A|. We s...
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