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For a natural number $Ngeq 2$ and a real $alpha$ such that $0 < alpha leq sqrt{N}-1$, we define $I_alpha:=[alpha,alpha+1]$ and $I_alpha^-:=[alpha,alpha+1)$ and investigate the continued fraction map $T_alpha:I_alpha to I_alpha^-$, which is defined as $T_alpha(x):= N/x-d(x),$ where $d(x):=left lfloor N/x -alpharight rfloor$. For all natural $N geq 7$, for certain values of $alpha$, open intervals $(a,b) subset I_alpha$ exist such that for almost every $x in I_{alpha}$ there is an natural number $n_0$ for which $T_alpha^n(x) otin (a,b)$ for all $ngeq n_0$. These emph{gaps} $(a,b)$ are investigated in the square $Upsilon_alpha:=I_alpha times I_alpha^-$, where the emph{orbits} $T_alpha^k(x), k=0,1,2,ldots$ of numbers $x in I_alpha$ are represented as cobwebs. The squares $Upsilon_alpha$ are the union of emph{fundamental regions}, which are related to the cylinder sets of the map $T_alpha$, according to the finitely many values of $d$ in $T_alpha$. In this paper some clear conditions are found under which $I_alpha$ is gapless. When $I_alpha$ consists of at least five cylinder sets, it is always gapless. In the case of four cylinder sets there are usually no gaps, except for the rare cases that there is one, very wide gap. Gaplessness in the case of two or three cylinder sets depends on the position of the endpoints of $I_alpha$ with regard to the fixed points of $I_alpha$ under $T_alpha$.
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A folklore conjecture in number theory states that the only integers whose expansions in base $3,4$ and $5$ contain solely binary digits are $0, 1$ and $82000$. In this paper, we present the first progress on this conjecture. Furthermore, we investigate the density of the integers containing only binary digits in their base $3$ or $4$ expansion, whereon an exciting transition in behaviour is observed. Our methods shed light on the reasons for this, and relate to several well-known questions, such as Grahams problem and a related conjecture of Pomerance. Finally, we generalise this setting and prove that the set of numbers in $[0, 1]$ who do not contain some digit in their $b$-expansion for all $b geq 3$ has zero Hausdorff dimension.
For regular continued fraction, if a real number $x$ and its rational approximation $p/q$ satisfying $|x-p/q|<1/q^2$, then, after deleting the last integer of the partial quotients of $p/q$, the sequence of the remaining partial quotients is a prefix of that of $x$. In this paper, we show that the situation is completely different if we consider the Hurwitz continued fraction expansions of a complex number and its rational approximations. More specifically, we consider the set $E(psi)$ of complex numbers which are well approximated with the given bound $psi$ and have quite different Hurwitz continued fraction expansions from that of their rational approximations. The Hausdorff and packing dimensions of such set are determined. It turns out that its packing dimension is always full for any given approximation bound $psi$ and its Hausdorff dimension is equal to that of the $psi$-approximable set $W(psi)$ of complex numbers. As a consequence, we also obtain an analogue of the classical Jarnik Theorem in real case.
We study the dynamics of the map endomorphism of N-dimensional projective space defined by f(X)=AX^d, where A is a matrix and d is at least 2. When d>N^2+N+1, we show that the critical height of such a morphism is comparable to its height in moduli space, confirming a case of a natural generalization of a conjecture of Silverman.
If $mathfrak{p} subseteq mathbb{Z}[zeta]$ is a prime ideal over $p$ in the $(p^d - 1)$th cyclotomic extension of $mathbb{Z}$, then every element $alpha$ of the completion $mathbb{Z}[zeta]_mathfrak{p}$ has a unique expansion as a power series in $p$ with coefficients in $mu_{p^d -1} cup {0}$ called the Teichmuller expansion of $alpha$ at $mathfrak{p}$. We observe three peculiar and seemingly unrelated patterns that frequently appear in the computation of Teichmuller expansions, then develop a unifying theory to explain these patterns in terms of the dynamics of an affine group action on $mathbb{Z}[zeta]$.
It was discovered some years ago that there exist non-integer real numbers $q>1$ for which only one sequence $(c_i)$ of integers $c_i in [0,q)$ satisfies the equality $sum_{i=1}^infty c_iq^{-i}=1$. The set of such univoque numbers has a rich topological structure, and its study revealed a number of unexpected connections with measure theory, fractals, ergodic theory and Diophantine approximation. In this paper we consider for each fixed $q>1$ the set $mathcal{U}_q$ of real numbers $x$ having a unique representation of the form $sum_{i=1}^infty c_iq^{-i}=x$ with integers $c_i$ belonging to $[0,q)$. We carry out a detailed topological study of these sets. For instance, we characterize their closures, and we determine those bases $q$ for which $mathcal{U}_q$ is closed or even a Cantor set. We also study the set $mathcal{U}_q$ consisting of all sequences $(c_i)$ of integers $c_i in [0,q)$ such that $sum_{i=1}^{infty} c_i q^{-i} in mathcal{U}_q$. We determine the numbers $r >1$ for which the map $q mapsto mathcal{U}_q$ (defined on $(1, infty)$) is constant in a neighborhood of $r$ and the numbers $q >1$ for which $mathcal{U}_q$ is a subshift or a subshift of finite type.
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