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Suppose we have $n$ irrational numbers $\{ x_1, x_2, \ldots, x_n \}$. For a generic set of such numbers, we have the well-known theorem that there exist infinitely many integers $q$ such that

$$ \max\{ \parallel q x_1 \parallel, \parallel q x_2 \parallel, \ldots, \parallel q x_n \parallel\} < q^{-1/n}.$$

Here $\parallel \cdot \parallel $ means the distance of the number to the closest integer.

The point is, could this theorem be strengthened if the $n$ numbers are somehow related? As an example, what if

$$ \{x_1, x_2, x_3, x_4 \} = \{ \sqrt{2}, \sqrt{3}, \sqrt{2+\sqrt{3}}, \sqrt{2-\sqrt{3}}\}?$$

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  • $\begingroup$ I'm not clear if you're using the term generic in a technical sense (that is you are saying that $(x_1,\ldots,x_n)$ belong to a dense $G_\delta$ set), or whether you just mean for an arbitrary $n$-tuple. The fact you quote is a corollary of the pigeonhole principle, so is true for all $n$-tuples. $\endgroup$
    – Anthony Quas
    Commented Mar 15, 2017 at 7:02
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    $\begingroup$ By Lindenstrauss' et Al. result towards the Littlewood conjecture, your inequality (in a sharper form) holds for much larger set than plain "generic" (namely, up to dimension $0$ subset), $\endgroup$
    – Asaf
    Commented Mar 15, 2017 at 7:03
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    $\begingroup$ You may be interested in a result by Cassels and SD about numbers from real cubic fields (in retrospect, one of the first instances of the interplay between dynamics and NT) - dx.doi.org/10.1098%2Frsta.1955.0010 $\endgroup$
    – Asaf
    Commented Mar 15, 2017 at 7:04
  • $\begingroup$ @AnthonyQuas Yes, I mean arbitrary actually. $\endgroup$
    – S. Kohn
    Commented Mar 15, 2017 at 12:43
  • $\begingroup$ I gave an apparently unsatisfying answer to this same question on Math StackExchange. See: math.stackexchange.com/questions/2186331/… $\endgroup$
    – O. S. Dawg
    Commented Mar 25, 2017 at 19:35

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