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Product sets cannot contain long arithmetic progressions

Tom 163 / 2014

Dmitrii Zhelezov Acta Arithmetica 163 (2014), 299-307 MSC: Primary 11B25. DOI: 10.4064/aa163-4-1

Streszczenie

Let $B$ be a set of complex numbers of size $n$. We prove that the length of the longest arithmetic progression contained in the product set $B.B = \{bb'\mid b, b' \in B\}$ cannot be greater than $O(\frac{n\log^2 n}{\log \log n})$ and present an example of a product set containing an arithmetic progression of length $\Omega(n \log n)$.For sets of complex numbers we obtain the upper bound $O(n^{3/2})$.

Autorzy

  • Dmitrii ZhelezovDepartment of Mathematical Sciences
    Chalmers University of Technology
    and
    University of Gothenburg
    41296 Gothenburg, Sweden
    e-mail

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