Expansions In Non-integer Bases Research Articles

Optimal expansions of Kakeya sequences

AbstractWe investigate optimal expansions of Kakeya sequences for the representation of real numbers. Expansions of Kakeya sequences generalize the expansions in non-integer bases and they display analogous redundancy phenomena. In this paper, we characterize optimal expansions of Kakeya sequences, and we provide conditions for the existence of unique expansions with respect to Kakeya sequences.

Univoque graphs for non-integer base expansions

Unique expansions in non-integer bases have been investigated in many papers during the last thirty years. They are often conveniently generated by labeled directed graphs. We give a precise description of the set of sequences generated by these graphs. This provides a geometric explanation of many former abstract results in this domain. Our results are illustrated by many examples.

Difference of Cantor sets and frequencies in Thue--Morse type sequences

In a recent paper, Baker and Kong have studied the Hausdorff dimension of the intersection of Cantor sets with their translations. We extend their results to more general Cantor sets. The proofs rely on the frequencies of digits in unique expansions in non-integer bases. In relation with this, we introduce a practical method to determine the frequency of any given finite block in Thue--Morse type sequences.

Digit frequencies and self-affine sets with non-empty interior

In this paper we study digit frequencies in the setting of expansions in non-integer bases, and self-affine sets with non-empty interior. Within expansions in non-integer bases we show that if $\unicode[STIX]{x1D6FD}\in (1,1.787\ldots )$ then every $x\in (0,1/(\unicode[STIX]{x1D6FD}-1))$ has a simply normal $\unicode[STIX]{x1D6FD}$-expansion. We also prove that if $\unicode[STIX]{x1D6FD}\in (1,(1+\sqrt{5})/2)$ then every $x\in (0,1/(\unicode[STIX]{x1D6FD}-1))$ has a $\unicode[STIX]{x1D6FD}$-expansion for which the digit frequency does not exist, and a $\unicode[STIX]{x1D6FD}$-expansion with limiting frequency of zeros $p$, where $p$ is any real number sufficiently close to $1/2$. For a class of planar self-affine sets we show that if the horizontal contraction lies in a certain parameter space and the vertical contractions are sufficiently close to $1$, then every non-trivial vertical fibre contains an interval. Our approach lends itself to explicit calculation and gives rise to new examples of self-affine sets with non-empty interior. One particular strength of our approach is that it allows for different rates of contraction in the vertical direction.

Notes on Some of Kahane’s Works: Ingham Type Theorems and Bernoulli Convolutions

We discuss the optimality of Kahane’s multidimensional Ingham type theorem and we establish a relationship between the absolute continuity of Bernoulli convolutions and some results on expansions in non-integer bases.

Multiple common expansions in non-integer bases

Multiple common expansions in non-integer bases

Dynamical properties of S-gap shifts and other shift spaces

We study the dynamical properties of certain shift spaces. To help study these properties we introduce two new classes of shifts, namely boundedly supermultiplicative (BSM) shifts and balanced shifts. It turns out that any almost specified shift is both BSM and balanced, and any balanced shift is BSM. However, as we will demonstrate, there are examples of shifts which are BSM but not balanced. We also study the measure theoretic properties of balanced shifts. We show that a shift space admits a Gibbs state if and only if it is balanced. Restricting ourselves to S-gap shifts, we relate certain dynamical properties of an S-gap shift to combinatorial properties from expansions in non-integer bases. This identification allows us to use the machinery from expansions in non-integer bases to give straightforward constructions of S-gap shifts with certain desirable properties. We show that for any q∈(0,1) there is an S-gap shift which has the specification property and entropy q. We also use this identification to address the question, for a given q∈(0,1), how many S-gap shifts exist with entropy q? For certain exceptional values of q there is a unique S-gap shift with this entropy.

Common expansions in noninteger bases

In this paper we study the existence of simultaneous representations of real numbers in bases p > q > 1 with the digit set A = {−m, . . . , 0, . . . ,m}. Among other results, we prove if m = 1 and q < 2, then there is a continuum of sequences (ci) ∈ A∞ satisfying ∑∞ i=1 ciq −i = ∑∞ i=1 cip −i. On the other hand, if m = 1 and q ≥ 2 + √ 2, then only the trivial sequence (ci) = 0∞ satis es the former equality.

Robot's hand and expansions in non-integer bases

Combinatorics We study a robot hand model in the framework of the theory of expansions in non-integer bases. We investigate the reachable workspace and we study some configurations enjoying form closure properties. Keywords: Robot hand, discrete control, expansions in non-integer bases, expansions in complex bases

Discrete spectra and Pisot numbers

By the m-spectrum of a real number q>1 we mean the set Y^m(q) of values p(q) where p runs over the height m polynomials with integer coefficients. These sets have been extensively investigated during the last fifty years because of their intimate connections with infinite Bernoulli convolutions, spectral properties of substitutive point sets and expansions in noninteger bases. We prove that Y^m(q) has an accumulation point if and only if q<m+1 and q is not a Pisot number. Consequently a number of related results on the distribution of points of this form are improved.

Robot's finger and expansions in non-integer bases

We study a robot finger model in the framework of the theory of expansions in non-integer bases. We investigate the reachable set and its closure. A control policy to get approximate reachability is also proposed.

Optimal expansions in non-integer bases

For a given positive integer m m , let A = { 0 , 1 , … , m } A=\{0,1,\ldots ,m\} and q ∈ ( m , m + 1 ) q \in (m,m+1) . A sequence ( c i ) = c 1 c 2 … (c_i)=c_1c_2 \ldots consisting of elements in A A is called an expansion of x x if ∑ i = 1 ∞ c i q − i = x \sum _{i=1}^{\infty } c_i q^{-i}=x . It is known that almost every x x belonging to the interval [ 0 , m / ( q − 1 ) ] [0,m/(q-1)] has uncountably many expansions. In this paper we study the existence of expansions ( d i ) (d_i) of x x satisfying the inequalities ∑ i = 1 n d i q − i ≥ ∑ i = 1 n c i q − i \sum _{i=1}^n d_iq^{-i} \ge \sum _{i=1}^n c_i q^{-i} , n = 1 , 2 , … , n=1,2,\ldots , for each expansion ( c i ) (c_i) of x x .

Generalized golden ratios of ternary alphabets

Expansions in noninteger bases often appear in number theory and probability theory, and they are closely connected to ergodic theory, measure theory and topology. For two-letter alphabets the golden ratio plays a special role: in smaller bases only trivial expansions are unique, whereas in greater bases there exist nontrivial unique expansions. In this paper we determine the corresponding critical bases for all three-letter alphabets and we establish the fractal nature of these bases in function of the alphabets.

Dimension of countable intersections of some sets arising in expansions in non-integer bases

Abstract in Undetermined We consider expansions of real numbers in non-integer bases. These expansions are generated by beta-shifts. We prove that some sets arising in metric number theory have the countable intersection property. This allows us to consider sets of reals that have common properties in a countable number of different (non-integer) bases. Some of the results are new even for integer bases.

Expansions in non-integer bases: Lower, middle and top orders

Let q ∈ ( 1 , 2 ) ; it is known that each x ∈ [ 0 , 1 / ( q − 1 ) ] has an expansion of the form x = ∑ n = 1 ∞ a n q − n with a n ∈ { 0 , 1 } . It was shown in [P. Erdős, I. Joó, V. Komornik, Characterization of the unique expansions 1 = ∑ i = 1 ∞ q − n i and related problems, Bull. Soc. Math. France 118 (1990) 377–390] that if q < ( 5 + 1 ) / 2 , then each x ∈ ( 0 , 1 / ( q − 1 ) ) has a continuum of such expansions; however, if q > ( 5 + 1 ) / 2 , then there exist infinitely many x having a unique expansion [P. Glendinning, N. Sidorov, Unique representations of real numbers in non-integer bases, Math. Res. Lett. 8 (2001) 535–543]. In the present paper we begin the study of parameters q for which there exists x having a fixed finite number m > 1 of expansions in base q. In particular, we show that if q < q 2 = 1.71 … , then each x has either 1 or infinitely many expansions, i.e., there are no such q in ( ( 5 + 1 ) / 2 , q 2 ) . On the other hand, for each m > 1 there exists γ m > 0 such that for any q ∈ ( 2 − γ m , 2 ) , there exists x which has exactly m expansions in base q.

On the number of unique expansions in non-integer bases

Let q > 1 be a real number and let m = m ( q ) be the largest integer smaller than q. It is well known that each number x ∈ J q : = [ 0 , ∑ i = 1 ∞ m q − i ] can be written as x = ∑ i = 1 ∞ c i q − i with integer coefficients 0 ⩽ c i < q . If q is a non-integer, then almost every x ∈ J q has continuum many expansions of this form. In this note we consider some properties of the set U q consisting of numbers x ∈ J q having a unique representation of this form. More specifically, we compare the size of the sets U q and U r for values q and r satisfying 1 < q < r and m ( q ) = m ( r ) .