**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 $\inf T> 1$?** 



Is the infimum of Salem numbers > 1?
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 $1/4$.]


