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We study aperiodic and periodic tilings induced by the Rauzy fractal and its subtiles associated to beta-substitutions related to the polynomial x 3 − ax 2 − bx − 1 for a ≥ b ≥ 1. In particular, we compute the corresponding boundary graphs, describing the adjacencies in the tilings. These graphs are a valuable tool for more advanced studies of the… (More)

Several cryptosystems rely on fast calculations of linear combinations in groups. One way to achieve this is to use joint signed binary digit expansions of small “weight.” We study two algorithms, one based on nonadjacent forms of the coefficients of the linear combination, the other based on a certain joint sparse form specifically adapted to… (More)

Shift radix systems form a collection of dynamical systems depending on a parameter r which varies in the d-dimensional real vector space. They generalize well-known numeration systems such as beta-expansions, expansions with respect to rational bases, and canonical number systems. Beta-numeration and canonical number systems are known to be intimately… (More)

Let Fq be a finite field and consider the polynomial ring Fq[X]. Let Q ∈ Fq[X]. A function f : Fq[X] → G, where G is a group, is called strongly Q-additive, if f (AQ + B) = f (A) + f (B) holds for all polynomials A, B ∈ Fq[X] with deg B < deg Q. We estimate Weyl Sums in Fq[X] restricted by Q-additive functions. In particular, for a certain character E we… (More)

Let o be the maximal order of a number field. Belcher showed in the 1970s that every algebraic integer in o is the sum of pairwise distinct units, if the unit equation u+v = 2 has a non-trivial solution u, v ∈ o *. We generalize this result and give applications to signed double-base digit expansions.

- Julien Bernat, Benoˆıt Loridant, Jörg Thuswaldner
- 2009

Let α = −2 + √ −1 be a root of the polynomial p(x) = x 2 + 4x + 5. It is well-known that the pair (α, {0, 1, 2, 3, 4}) forms a canonical number system, i.e., that each γ ∈ Z[α] admits a finite representation of the shape γ = a 0 + a 1 α + · · · + a ℓ α ℓ with a i ∈ {0, 1, 2, 3, 4}. The set T of points with integer part 0 in this number system T := ∞ i=1 a i… (More)

Let k = Q(√ −D) be an imaginary quadratic number field with ring of integers Z k and let k(α) be the cubic extension of k generated by the polynomial f t (x) = x 3 − (t − 1)x 2 − (t + 2)x − 1 with t ∈ Z k. In the present paper we characterize all elements γ ∈ Z k [α] with norms satisfying |N k(α)/k | ≤ |2t + 1| for |t| ≥ 14. This generalizes a corresponding… (More)

- Yu-Long Deng, Caihong Hu, Shunchao Long, Tai-Man Tang, Jörg Thuswaldner, Lifeng Xi
- 2013

We prove that the sharp lower bounds of the Minkowski and Hausdorff dimensions of circular Kakeya sets in R are 1/2 and 0 respectively .

Let Fq be a finite field with q elements and p ∈ Fq[X, Y ]. In this paper we study properties of additive functions with respect to number systems which are defined in the ring Fq[X, Y ]/p Fq[X, Y ]. Our results comprise distribution results, exponential sum estimations as well as a version of Waring's Problem restricted by such additive functions. Similar… (More)