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2 results

  • MONOTONIC PHINOMIAL COEFFICIENTS

  • Part of
  • FLORIAN LUCA, PANTELIMON STĂNICĂ
  • Journal:
    Bulletin of the Australian Mathematical Society / Volume 95 / Issue 3 / June 2017
    Published online by Cambridge University Press:
    09 January 2017, pp. 365-372
    Print publication:
    June 2017
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  • We investigate the monotonic characteristics of the generalised binomial coefficients (phinomials) based upon Euler’s totient function. We show, unconditionally, that the set of integers for which this sequence is unimodal is finite and, assuming the generalised Riemann hypothesis, we find all the exceptions.

  • Arithmetic Progressions in Sumsets and Lp-Almost-Periodicity

  • ERNIE CROOT, IZABELLA ŁABA, OLOF SISASK
  • Journal:
    Combinatorics, Probability and Computing / Volume 22 / Issue 3 / May 2013
    Published online by Cambridge University Press:
    19 March 2013, pp. 351-365

  • We prove results about the Lp-almost-periodicity of convolutions. One of these follows from a simple but rather general lemma about approximating a sum of functions in Lp, and gives a very short proof of a theorem of Green that if A and B are subsets of {1,. . .,N} of sizes αN and βN then A+B contains an arithmetic progression of length at least

    \begin{equation} \exp ( c (\alpha \beta \log N)^{1/2} - \log\log N). \end{equation}
    Another almost-periodicity result improves this bound for densities decreasing with N: we show that under the above hypotheses the sumset A+B contains an arithmetic progression of length at least
    \begin{equation} \exp\biggl( c \biggl(\frac{\alpha \log N}{\log^3 2\beta^{-1}} \biggr)^{1/2} - \log( \beta^{-1} \log N) \biggr). \end{equation}