Salem numbers, Pisot numbers, Mahler measure and graphs

短Salem多项式

Short Salem polynomials

本文对长度为5的Salem多项式进行了完整分类，并证明长度为6的Salem多项式除了有限多个外均属于12个无限族，基于Lehmer猜想给出了126个例外情况的完整列表。

On Salem numbers, expansive polynomials and Stieltjes continued fractions

A converse method to the Construction of Salem (1945) of convergent families of Salem numbers is investigated in terms of an association between Salem polynomials and Hurwitz quotients via expansive polynomials of small Mahler measure. This association makes use of Bertin-Boyd's Theorem A (1995) of interlacing of conjugates on the unit circle; in this context, a Salem number $β$ is produced and coded by an m-tuple of positive rational numbers characterizing the (SITZ) Stieltjes continued fraction of the corresponding Hurwitz quotient (alternant). The subset of Stieltjes continued fractions ove…

帕里数的帕里阶数

Parry order of Parry numbers

本文引入帕里阶数，研究帕里数的性质，并证明帕里数的帕里阶数与Pisot和Salem数相关，同时给出帕里阶数的上界和实例。

On the co-factors of degree 6 Salem number beta expansions

For $β> 1$, a sequence $(c_n)_{n \geq 1} \in \mathbb{Z}^{\mathbb{N}^+}$ with $0 \leq c_n < β$ is the \emph{beta expansion} of $x$ with respect to $β$ if $x = \sum_{n = 1}^\infty c_nβ^{-n}$. Defining $d_β(x)$ to be the greedy beta expansion of $x$ with respect to $β$, it is known that $d_β(1)$ is eventually periodic as long as $β$ is a Pisot number. It is conjectured that the same is true for Salem numbers, but is only currently known to be true for Salem numbers of degree 4. Heuristic arguments suggest that almost all degree 6 Salem numbers admit periodic expansions but that a positive proport…

On the rational approximations to the powers of an algebraic number

About fifty years ago Mahler proved that if $α>1$ is rational but not an integer and if $0<l<1$ then the fractional part of $α^n$ is $>l^n$ apart from a finite set of integers $n$ depending on $α$ and $l$. Answering completely a question of Mahler we show that the same conclusion holds for all algebraic numbers which are not $d$-th roots of Pisot numbers. By related methods, we also answer a question by Mendes France, characterizing completely the quadratic irrationals $α$ such that the continued fraction of $α^n$ has period length tending to infinity.

单生严格佩龙多项式

Monogenic Strictly-Perron Polynomials

本文明确单生多项式和严格佩龙多项式的定义，并证明对任意$n\ge 2$的自然数，存在无穷多$n$次单生严格佩龙多项式。

On periodic representations in non-Pisot bases

We study periodic expansions in positional number systems with a base $β\in\C,\ |β|>1$, and with coefficients in a finite set of digits $\A\subset\C.$ We are interested in determining those algebraic bases for which there exists $\A\subset \Q(β),$ such that all elements of $\Q(β)$ admit at least one eventually periodic representation with digits in $\A$. In this paper we prove a general result that guarantees the existence of such an $\A$. This result implies the existence of such an $\A$ when $β$ is a rational number or an algebraic integer with no conjugates of modulus $1$. We also consider …

Arithmetics in number systems with negative base

We study the numeration system with negative basis, introduced by Ito and Sadahiro. We focus on arithmetic operations in the set ${\rm Fin}(-β)$ and $\Z_{-β}$ of numbers having finite resp. integer $(-β)$-expansions. We show that ${\rm Fin}(-β)$ is trivial if $β$ is smaller than the golden ratio $\frac12(1+\sqrt5)$. For $β\geq\frac12(1+\sqrt5)$ we prove that ${\rm Fin}(-β)$ is a ring, only if $β$ is a Pisot or Salem number with no negative conjugates. We prove the conjecture of Ito and Sadahiro that ${\rm Fin}(-β)$ is a ring if $β$ is a quadratic Pisot number with positive conjugate. For quadr…

A Proof of the Conjecture of Lehmer and of the Conjecture of Schinzel-Zassenhaus

The conjecture of Lehmer is proved to be true. The proof mainly relies upon: (i) the properties of the Parry Upper functions $f_{house(α)}(z)$ associated with the dynamical zeta functions $ζ_{house(α)}(z)$ of the Rényi--Parry arithmetical dynamical systems, for $α$ an algebraic integer $α$ of house "$house(α)$" greater than 1, (ii) the discovery of lenticuli of poles of $ζ_{house(α)}(z)$ which uniformly equidistribute at the limit on a limit "lenticular" arc of the unit circle, when $house(α)$ tends to $1^+$, giving rise to a continuous lenticular minorant ${\rm M}_{r}(house(α))$ of the Mahler…

A proof of the Conjecture of Lehmer

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 …

伯努利卷积——2023

Bernoulli convolutions -- 2023

本文研究参数在(1,2)内的伯努利卷积测度，证明代数非Pisot数的维数为1、Salem数情形下测度等价于勒贝格测度且密度无界，并给出De-Jun Feng定理的简短证明。

十七阶多对数阶梯

A seventeenth-order polylogarithm ladder

本文通过加入指标630，在已知四个16阶多对数阶梯基础上，构造了唯一的17阶阶梯，并验证了其涉及125个常数、近300位整数系数的整数关系，精度超过59000位小数。