Yann Bugeaud

Classical and modular approaches to exponential Diophantine equations II. The Lebesgue–Nagell equation

We solve completely the Lebesgue-Nagell equation x^2+D=y^n, in integers x, y, n>2, for D in the range 1 =< D =< 100.

ON THE DISTANCE BETWEEN ROOTS OF INTEGER POLYNOMIALS

We study families of integer polynomials having roots very close to each other.

POLYNOMIAL ROOT SEPARATION

We discuss the following question: How close to each other can two distinct roots of an integer polynomial be? We summarize what is presently known on this and related problems, and establish several new results on root separation of monic, integer polynomials.

Root separation for irreducible integer polynomials

We establish new results on root separation of integer, irreducible polynomials of degree at least four. These improve earlier bounds of Bugeaud and Mignotte (for even degree) and of Beresnevich, Bernik, and Goetze (for odd degree).

On shrinking targets for Zm actions on tori

Let A be an n by m matrix with real entries. Consider the set Bad_A of x \in [0,1)^n for which there exists a constant c(x)>0 such that for any q \in Z^m the distance between x and the point {Aq} is at least c(x) |q|^{-m/n}. It is shown that the intersection of Bad_A with any suitably regular fractal set is of maximal Hausdorff dimension. The linear form systems investigated in this paper are natural extensions of irrational rotations of the circle. Even in the latter one-dimensional case, the results obtained are new.

Distribution of full cylinders and the Diophantine properties of the orbits in

Letˇ > 1 be a real number. LetTˇ denote theˇ-transformation on Œ0; 1 . A cylinder of order n is a set of real numbers in Œ0; 1 having the same first n digits in their ˇ-expansion. A cylinder is called full if it has maximal length, i.e., if its length is equal to ˇ . In this paper, we show that full cylinders are well distributed in Œ0; 1 in a suitable sense. As an application to the metrical theory of ˇ-expansions, we determine the Hausdorff dimension of the set fx 2 Œ0; 1 W jT n ˇ x znj < e Snf .x/ for infinitely many n 2 Ng; where fzngn 1 is a sequence of real numbers in Œ0; 1 , the function f W Œ0; 1 ! R is continuous, and Snf .x/ denotes the ergodic sum f .x/C C f .T n 1 ˇ x/. Mathematics Subject Classification (2010). 11K55, 28A80.

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For a positive integer

-expansions

n

On simultaneous rational approximation to a real number and its integral powers

and a real number

On a problem of Diophantus for higher powers

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