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  • Author or Editor: Imre Z. Ruzsa x
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Concluding remark It would be interesting to know whether it is possible to generalize the method of the previous section to all commutative locally compact groups. For a long time I tried in vain to do this. Mauclaire uses a different approach. He applies the structure theory of locally compact groups to reduce the general case to certain subcases, which were studied in this paper.

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It is a classical unsolved problem whether there is a polynomial with integral coef- ficients whose values at natural numbers form a Sidon set. In this note we prove the existence of a polynomial of degree 5, with real coeficients, such that the integer parts of the values form a Sidon set.

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Summary  

Inspired by a paper of Srkzy [4] we study sets of integers and sets of residues with the property that all sums and all products are distinct.

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Let A be a set of n real numbers such that the number of distinct twofold sums is a n. We show that the number of twofold products is = c n 2/ (a4 log n), and the number of quotients is = c n 2/ min (a6, a4 log n) with some absolute constant c. For bounded a this gives the correct order of magnitude for the quotients. For sums we think that the correct order is n2/ (log n)a with some a<1, perhaps with 2 log 2 -1, as a result of Pomerance and Sárközy suggests. We also give more general inequalities for sums, products and quotients formed with different sets. The proofs use geometric tools, mainly the Szemerédi-Trotter inequality.

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