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Register a new userCollusions in Teichmuller expansions
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Authors
Trevor Hyde
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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 $mathbb{Z}[zeta]$.
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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 $T_alpha$.
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.
A new model of collusions in an organization is proposed. Each actor $a_{i=1,cdots,N}$ disposes one unique good $g_{j=1,cdots,N}$. Each actor $a_i$ has also a list of other goods which he/she needs, in order from desired most to those desired less. Finally, each actor $a_i$ has also a list of other agents, initially ordered at random. The order in the last list means the order of the access of the actors to the good $g_j$. A pair after a pair of agents tries to make a transaction. This transaction is possible if each of two actors can be shifted upwards in the list of actors possessed by the partner. Our numerical results indicate, that the average time of evolution scales with the number $N$ of actors approximately as $N^{2.9}$. For each actor, we calculate the Kendalls rank correlation between the order of desired goods and actors place at the lists of the goods possessors. We also calculate individual utility funcions $eta_i$, where goods are weighted according to how strongly they are desired by an actor $a_i$, and how easily they can be accessed by $a_i$. Although the individual utility functions can increase or decrease in the time course, its value averaged over actors and independent simulations does increase in time. This means that the system of collusions is profitable for the members of the organization.
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.
A beta expansion is the analogue of the base 10 representation of a real number, where the base may be a non-integer. Although the greedy beta expansion of 1 using a non-integer base is in general infinitely long and non-repeating, it is known that if the base is a Pisot number, then this expansion will always be finite or periodic. Some work has been done to learn more about these expansions, but in general these expansions were not explicitly known. In this paper, we present a complete list of the greedy beta expansions of 1 where the base is any regular Pisot number less than 2, revealing a variety of remarkable patterns. We also answer a conjecture of Boyds regarding cyclotomic co-factors for greedy expansions.
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