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BACKGROUND

A Salem number is an algebraic integer $\theta$ such that all the Galois conjugates of $\theta$ are $\leq 1$ in absolute value, and at least one of them lies on the unit circle. Their importance is derived for example from the fact that the minimal polynomial of a Salem number, $$ P(x) = x^{10} + x^9 - x^7 - x^6 - x^5 - x^4 - x^3 + x + 1 $$ is conjectured to minimize the Mahler measure ($M(P) = 1.17...$) over all $P \in \mathbb{Z}[x]$ with $M(P) > 1$.

The closely related Pisot numbers are algebraic integers $\theta > 1$ such that all the Galois conjugates of $\theta$ are of absolute value $< 1$. Their set is closed and in particular there exists a smallest Pisot number. This has been found by Siegel to be the plastic constant, $\theta_0 = 1,32471\ldots$, a root of $g(x) = x^3 - x - 1$. It is known that for any monic, non-reciprocal polynomial $P$ we have $M(P) \geq M(g) = \theta_0$. This is Smyth's theorem.

THE QUESTION

I am currently reading ``Conjecture de Lehmer et petits nombres de Salem" by Bertin and Pathiaux-Delefosse. The book is from 1989 and it is stated in it that it is still not known if $\inf T > 1$ where $T$ is the set of all Salem numbers. Has there been any developments since then? Is this conjecture still open?

Precisely: Has it been proved or disproved that $\inf T> 1$?

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  • $\begingroup$ I am aware that the title of this question is not the most fortunate one, but I think that it captures the gist of the question rather succintly. $\endgroup$
    – blabler
    Commented Jul 25, 2013 at 21:29
  • $\begingroup$ Yes according to wikipedia en.wikipedia.org/wiki/Lehmer%27s_conjecture $\endgroup$
    – Anthony Quas
    Commented Jul 25, 2013 at 21:34
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    $\begingroup$ Obviously it has not been disproved that $\inf T>1$. Otherwise Lehmer's conjecture would be known to be false. Further, if you look at the (reasonably authoritative wiki page) you will see that there are bounds on the Mahler measure which converge to 0 in the degree. These bounds would be rendered moot if there were a lower bound independent of the degree (i.e. $\inf T > 1$). Hence, assuming that the editors of the wiki page did not fall asleep for the last few years, one can reasonably conclude that it has neither been proved nor disproved that $\inf T>1$. $\endgroup$
    – Anthony Quas
    Commented Jul 25, 2013 at 23:11
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    $\begingroup$ Historical note --- Lehmer would remind people that he had not published it as a conjecture, only a question, since he didn't feel he had enough evidence for it to call it a conjecture. $\endgroup$
    – Gerry Myerson
    Commented Jul 25, 2013 at 23:54
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    $\begingroup$ Some progress on the Salem number conjecture: ams.org/mathscinet-getitem?mr=1824892 ams.org/mathscinet-getitem?mr=1953192 ams.org/mathscinet-getitem?mr=2105815 $\endgroup$
    – Ian Agol
    Commented Jul 26, 2013 at 3:49

3 Answers 3

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I believe it is the general opinion, at least among those working in diophantine approximations, that the extreme case of Salem numbers (the question of the title) would be just as difficult as the full Lehmer conjecture. It is not a coincidence that the smallest known Mahler measures are realized by Salem numbers, and I have not heard of any improvement on the general Dobrowolski bound $\log{M(P)} > \big(\frac{9}{4} - o(1) \big) \Big( \frac{\log{\log{d}}}{\log{d}} \Big)^3$ under restricting to the Salem case.

[The constant $9/4$, due to Loubotin in 1983, is apparently the best that Dobrowolski's method can produce. Voutier has shown that the inequality holds without exception with the constant $1/4$.]

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According to [1(1992), 2(2011)] it is not known if $T$ is dense in $[1,\infty)$. Therefore it has not been proved that $\inf T>1$.
Since, according to wikipedia, Lehmer's problem is still open it has not been disproved either.

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    $\begingroup$ Your answer is just as good as the other answer. I had to make a pick which one to accept. $\endgroup$
    – blabler
    Commented Nov 23, 2013 at 23:43
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Just wanted to add: the conjecture that Salem numbers are bounded away from one (the "Salem Conjecture") is wide open as the other answers state, and it's equivalent to part of what is known as the "Short Geodesic Conjecture" for hyperbolic orbifolds: that there is a positive universal lower bound for the lengths of geodesics in arithmetic hyperbolic 2-orbifolds. The case of 3-orbifolds implies the Salem Conjecture. You can find this information in Maclachlan and Reid's book The Arithmetic of Hyperbolic 3-Manifolds, Section 12.3, along with some comments on the Salem Conjecture.

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