Let g be a nonconstant rational map from the projective line to itself that has degree greater than one and is defined over a number field. The map g gives rise to generalized Mahler measures for polynomials in one variable. We use diophantine approximation to show that the generalized Mahler measure of a polynomial F at a place v can be computed by averaging the log of the v-adic absolute value of F over the periodic points of g. This allows us to compute canonical heights of algebraic points via equidistribution on periodic points.
Let X be a projective integral normal scheme over a number field F; let L be a ample line bundle on X together with a semi-positive adelic metric in the sense of Zhang. The main results of this article are 1) A formula which computes the height (relative to L) of a Cartier divisor D on X as generalized "Mahler measures", i.e. by integrating Green functions for D against measures attached to L. 2) a theorem of equidistribution of points of "small" height valid for functions with logarithmic singularities along a divisor D, provided the height of D is "minimal". In the context of algebraic dynam…
We study the dynamics of stochastic families of rational maps on the projective line. As such families can be infinite and may not typically be defined over a single number field, we introduce the concept of generalized adelic measures, generalizing previous notions introduced by Favre and Rivera-Letelier and Mavraki and Ye. Generalized adelic measures are defined over the measure space of places of an algebraic closure of the rationals, using a framework established by Allcock and Vaaler. This turns our heights from sums over places into integrals. We prove an equidistribution result for gene…
For the generalized Dedekind sums s_{ij}(p,q) defined in association with the x^{i}y^{j}-coefficient of the Todd power series of the lattice cone in R^2 generated by (1,0) and (q,p), we associate an exponential sum. We obtain this exponential sum using the cocycle property of the Todd series of 2d cones and the nonsingular cone decomposition along with the continued fraction of q/p. Its Weil bound is given for the modulus q, applying the purity theorem of the cohomology of the related l-adic sheaf due to Denef and Loeser. The Weil type bound of Denef and Loeser fulfills the Weyl's equidistribu…
In this work we apply the techniques that were developed in [Lalin: An algebraic integration for Mahler measure] in order to study several examples of multivariable polynomials whose Mahler measure is expressed in terms of special values of the Riemann zeta function or Dirichlet L-series. The examples may be understood in terms of evaluations of regulators. Moreover, we apply the same techniques to the computation of generalized Mahler measures, in the sense of Gon and Oyanagi [Gon, Oyanagi: Generalized Mahler measures and multiple sine functions]
We prove a variety of explicit formulas relating special values of generalized hypergeometric functions to lattice sums with four indices of summation. These results are related to Boyd's conjectured identities between Mahler measures and special values of $L$-series of elliptic curves.
The metric Mahler measure was first studied by Dubickas and Smyth in 2001 as a means of phrasing Lehmer's conjecture in topological language. More recent work of the author examined a parametrized family of generalized metric Mahler measures that gives rise to a series of new, and apparently difficult, problems. We establish a connection between these metric Mahler measures and the theory of continued fractions in a certain class of special cases. Our results enable us to calculate metric Mahler measures in several new examples.
The Conjecture of Lehmer is proved to be true. The proof mainly relies upon: (i) the properties of the Parry Upper functions $f_{\houseα}(z)$ associated with the dynamical zeta functions $ζ_{\houseα}(z)$ of the Rényi--Parry arithmetical dynamical systems ($β$-shift), for $α$ a reciprocal algebraic integer of house $\houseα$ greater than 1, (ii) the discovery of lenticuli of poles of $ζ_{\houseα}(z)$ which uniformly equidistribute at the limit on a limit "lenticular" arc of the unit circle, when $\houseα$ tends to $1^+$, giving rise to a continuous lenticular minorant ${\rm M}_{r}(\houseα)$ of …
The Mahler measure of a nonzero $n$-variable polynomial $P$ is the integral of $\log|P|$ on the unit $n$-torus. A result of Boyd and Lawton says that the Mahler measure of a multivariate polynomial is the limit of Mahler measures of univariate polynomials. We prove the analogous result for different extensions of Mahler measure such as generalized Mahler measure (integrating the maximum of $\log|P|$ for possibly different $P$'s), multiple Mahler measure (involving products of $\log|P|$ for possibly different $P$'s), and higher Mahler measure (involving $\log^k|P|$).
This work addresses problems on simultaneous Diophantine approximation on fractals, motivated by a long standing problem of Mahler regarding Cantor's middle $1/3$ set. We obtain the first instances where a complete analogue of Khintchine's Theorem holds for fractal measures. Our results apply to fractals which are self-similar by a system of rational similarities of $\mathbb{R}^d$ (for any $d\geq 1$) and have sufficiently small Hausdorff co-dimension. A concrete example of such measures in the context of Mahler's problem is the Hausdorff measure on the "middle $1/5$ Cantor set"; i.e. the set o…
In the first part we construct algorithms which we apply to solve S-unit, Mordell, cubic Thue, cubic Thue-Mahler and generalized Ramanujan-Nagell equations. As a byproduct we obtain alternative practical approaches for various classical Diophantine problems, including the fundamental problem of finding all elliptic curves over Q with good reduction outside a given finite set of rational primes. To illustrate the utility of our algorithms we determined the solutions of large classes of equations, containing many examples of interest which are out of reach for the known methods. In addition we u…
For an algebraic number $α$, the metric Mahler measure $m_1(α)$ was first studied by Dubickas and Smyth in 2001 and was later generalized to the $t$-metric Mahler measure $m_t(α)$ by the author in 2010. The definition of $m_t(α)$ involves taking an infimum over a certain collection $N$-tuples of points in $\overline{\mathbb Q}$, and from previous work of Jankauskas and the author, the infimum in the definition of $m_t(α)$ is attained by rational points when $α\in \mathbb Q$. As a consequence of our main theorem in this article, we obtain an analog of this result when $\mathbb Q$ is replaced wi…
Let f be a generalized modular function of weight 0 of level N such that its q-exponents c(n)(n>0) are all real, and div(f) is zero. In this note, we show the equidistribution of signs for c(p)(p prime) by using equidistribution theorems for normalized cuspidal eigenforms of integral weight.
This paper proves a commutative algebraic extension of a generalized Skolem-Mahler-Lech theorem due to the first author. Let $A$ be a finitely generated commutative $K$-algebra over a field of characteristic $0$, and let $σ$ be a $K$-algebra automorphism of $A$. Given ideals $I$ and $J$ of $A$, we show that the set $S$ of integers $m$ such that $σ^m(I) \supseteq J$ is a finite union of complete doubly infinite arithmetic progressions in $m$, up to the addition of a finite set. Alternatively, this result states that for an affine scheme $X$ of finite type over $K$, an automorphism $σ\in {\rm Au…
We present a quantitative version of Bilu's theorem on the limit distribution of Galois orbits of sequences of points of small height in the $N$-dimensional algebraic torus. Our result gives, for a given point, an explicit bound for the discrepancy between its Galois orbit and the uniform distribution on the compact subtorus, in terms of the height and the generalized degree of the point.
We consider a family of heights defined by the $L_p$ norms of polynomials with respect to the equilibrium measure of a lemniscate for $0 \le p \le \infty$, where $p=0$ corresponds to the geometric mean (the generalized Mahler measure) and $p=\infty$ corresponds to the standard supremum norm. This special choice of the measure allows to find an explicit form for the geometric mean of a polynomial, and estimate it via certain resultant. For lemniscates satisfying appropriate hypotheses, we establish explicit polynomials of lowest height, and also show their uniqueness. We discuss relations betwe…
We establish the equidistribution with respect to the bifurcation measure of post-critically finite maps in any one-dimensional algebraic family of unicritical polynomials. Using this equidistribution result, together with a combinatorial analysis of certain algebraic correspondences on the complement of the Mandelbrot set $M_2$ (or generalized Mandelbrot set $M_d$ for degree $d>2$), we classify all complex plane curves $C$ with Zariski-dense subsets of points $(a,b)\in C$, such that both $z^d+a$ and $z^d+b$ are simultaneously post-critically finite for a fixed degree $d\geq 2$. Our result is …
Mahler's measure is generalized to create the class of {\it multiplicative distance functions}. These functions measure the complexity of polynomials based on the location of their zeros in the complex plane. Following work of S.-J. Chern and J. Vaaler in \cite{chern-vaaler}, we associate to each multiplicative distance function two families of analytic functions which encode information about its range on \C[x] and \R[x]. These {\it moment functions} are Mellin transforms of distribution functions associated to the multiplicative distance function and demonstrate a great deal of arithmetic st…
