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Fix a $B < \infty$. How small can the infimum be, as a function of $m$, of the difference $|\alpha - \beta|$ of a pair of distinct roots of an integer polynomial of degree $m$ having Mahler measure bounded by $B$?

My guess was that this difference cannot get smaller than $O_B(1/m)$, but I was wrong that this follows from an effective equidistribution theorem. I do not know about any literature on this question. I only needed this with $\beta = \zeta_n$ a root of unity of order $n \asymp m$, but the available lower bounds on $|\alpha^n - 1|$ do not seem to be of any help for this. If my hope that $|\alpha - \beta| > c(B)/m$ were correct, the subexponential bound would follow by the argument left below.

Fix a $L < \infty$. How small can the infimum be, as a function of $m$, of the difference $|\alpha - \beta|$ of a pair of distinct roots of an integer polynomial of degree $m$ having length bounded by $L$?

My guess was that this difference cannot get smaller than $O_L(1/m)$, but I was wrong that this follows from an effective equidistribution theorem. I do not know about any literature on this question. I only needed this with $\beta = \zeta_n$ a root of unity of order $n \asymp m$, but the available lower bounds on $|\alpha^n - 1|$ do not seem to be of any help for this. If my hope that $|\alpha - \beta| > c(L)/m$ were correct, the subexponential bound would follow by the argument left below.

• it has a Lipschitz constant bounded by $10n/C$;
• it is bounded in magnitude by $\log{(n/C)}$ in $D(n,C)$, and by a constant independent of $n$ in $\mathbb{C} \setminus K$;
• its Laplacian is bounded in magnitude by $10(n/C)^2$ in $D(n,C)$ and by a constant independent of $n$ in $\mathbb{C} \setminus K$.

• it has a Lipschitz constant bounded by $10n/C$;
• it is bounded in magnitude by $10\log{(n/C)}$ in $D(n,C)$, and by a constant independent of $n$ in $\mathbb{C} \setminus K$;
• its Laplacian is bounded in magnitude by $10(n/C)^2$ in $D(n,C)$ and by a constant independent of $n$ in $\mathbb{C} \setminus K$.

To prove our bound we will need the quantitative equidistribution theorem of Favre and Rivera-Letelier. The following variant is stated on page 3 of this paper (On totally real numbers and equidistribution) by Fili and Miner: http://arxiv.org/pdf/1210.7887.pdf . The only modification I make to the statement below is that the polynomial $P$ is not required to be irreducible, but irreducibility is irrelevant in such statements.

I had thought this had the following as consequence. It certainly holds for $\delta$ bounded away from zero, but my intended application had to have $\delta$ as small as $O(1/m)$; then, as Terry Tao points out in the comments, the first term on the right-hand side of the estimate in the above theorem, involving the Lipschitz constant, destroys all hope. (The Laplacian under the integral poses no problem on the other hand, but I had overlooked the Lipschitz constant.)

The following is a variant of the quantitative equidistribution theorem of Favre and Rivera-Letelier, taken from page 3 of this paper (On totally real numbers and equidistribution) by Fili and Miner: http://arxiv.org/pdf/1210.7887.pdf . The only modification I make to the statement is that the polynomial $P$ is not required to be irreducible; but irreducibility is irrelevant in such statements.

I had thought this had the following as consequence. It certainly holds for $\delta$ bounded away from zero, but my intended application had to have $\delta$ as small as $O(1/m)$; then the first term on the right-hand side of the estimate destroys all hope, as Terry Tao noted in the comments. The Laplacian under the integral poses no problem, but I had overlooked the Lipschitz constant.