On the Lehmer conjecture and counting in finite fields

On the Lehmer conjecture and counting in finite fields, Discrete Analysis 2019:5, 8pp.

Let $\alpha$ be an algebraic number and let $a_0\prod_{i=1}^k(x-\alpha_i)$ be its minimal polynomial. The Mahler measure of $\alpha$ is defined to be the quantity $|a_0|\prod_{i=1}^k\max\{1,|\alpha_i|\}$. For example, $\sqrt 2$ has minimal polynomial $x^2-2=(x-\sqrt 2)(x+\sqrt 2)$, so its Mahler measure is $2$, and if $\alpha$ is a root of unity, then its Mahler measure is 1. The Lehmer conjecture asserts that there exists an absolute constant $c_0>0$ such that every algebraic number that is not a root of unity has Mahler measure at least $1+c_0$. The smallest known Mahler measure greater than 1 comes from the polynomial
$$x^{10}+x^9-x^7-x^6-x^5-x^4-x^3+x+1,$$
which has exactly one root outside the unit disc (making that root a so-called Salem number), which is approximately equal to $1.176280818$. Thus, $0.176280819$ is an upper bound for $c_0$, if it exists.

Let $S_d$ be the set of all polynomials of degree at most $d$ with coefficients in $\{0,1\}$. In an earlier paper [1], the authors of this paper proved that the Lehmer conjecture is equivalent to the assertion that the growth rate of the set $\{P(\alpha):P\in S_d\}$ as $d$ tends to infinity is at least $(1+c_1)^d$ for some absolute constant $c_1>0$, again as long as $\alpha$ is not a root of unity. (If it is a root of unity, then the growth rate is polynomial.)

The purpose of this short paper is to present a further equivalence. The authors define a prime $p$ to be $C$-wild if there exists some $x$ of multiplicative order at least $(\log p)^2$ such that the $(\log p)^C$-fold sumset of the geometric progression $H=\{x,x^2,\dots,x^{|C\log p|}\}$ is not the whole of $\mathbb F_p$. They then prove that the Lehmer conjecture is true if and only if there exists $C$ such that the proportion of primes less than $n$ that are $C$-wild tends to zero as $n$ tends to infinity.

The “if” direction of this equivalence turns out not to be too hard: with the help of well-known results, an algebraic number $\alpha$ with Mahler measure close to 1 can be used to show that there is a constant $C$ and a dense set of $C$-wild primes. For this reason, the authors do not claim that counting wild primes is likely to be easier than proving the Lehmer conjecture directly. The fact that the Lehmer conjecture is related to additive combinatorics has been observed already, as the authors acknowledge, but the connection is particularly cleanly expressed here. The principal novelty of the paper is the converse statement. It is quite surprising that a well-known conjecture in algebraic number theory is actually equivalent to a simple counting problem in $\mathbb F_p$: this sheds new light on the conjecture and perhaps helps to explain why it is difficult.

[1] Breuillard, E. and Varjú, P. P., Entropy of Bernoulli convolutions and uniform exponential growth for linear groups, J. Anal. Math., to appear. Also available at arXiv:1510.04043