**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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