Patterns in rational base number systems

Number systems with a rational number $a/b > 1$ as base have gained interest in recent years. In particular, relations to Mahler's 3/2-problem as well as the Josephus problem have been established. In the present paper we show that the patterns of digits in the representations of positive integers in such a number system are uniformly distributed. We study the sum-of-digits function of number systems with rational base $a/b$ and use representations w.r.t. this base to construct normal numbers in base $a$ in the spirit of Champernowne. The main challenge in our proofs comes from the fact that the language of the representations of integers in these number systems is not context-free. The intricacy of this language makes it impossible to prove our results along classical lines. In particular, we use self-affine tiles that are defined in certain subrings of the ad\'ele ring $\mathbb{A}_\mathbb{Q}$ and Fourier analysis in $\mathbb{A}_\mathbb{Q}$. With help of these tools we are able to reformulate our results as estimation problems for character sums

Oligomeric Status and Nucleotide Binding Properties of the Plastid ATP/ADP Transporter 1: Toward a Molecular Understanding of the Transport Mechanism

Background: Chloroplast ATP/ADP transporters are essential to energy homeostasis in plant cells. However, their molecular mechanism remains poorly understood, primarily due to the difficulty of producing and purifying functional recombinant forms of these transporters. Methodology/Principal Findings: In this work, we describe an expression and purification protocol providing good yields and efficient solubilization of NTT1 protein from Arabidopsis thaliana. By biochemical and biophysical analyses, we identified the best detergent for solubilization and purification of functional proteins, LAPAO. Purified NTT1 was found to accumulate as two independent pools of well folded, stable monomers and dimers. ATP and ADP binding properties were determined, and Pi, a co-substrate of ADP, was confirmed to be essential for nucleotide steady-state transport. Nucleotide binding studies and analysis of NTT1 mutants lead us to suggest the existence of two distinct and probably inter-dependent binding sites. Finally, fusion and deletion experiments demonstrated that the C-terminus of NTT1 is not essential for multimerization, but probably plays a regulatory role, controlling the nucleotide exchange rate. Conclusions/Significance: Taken together, these data provide a comprehensive molecular characterization of a chloroplas

30 years of collaboration

We highlight some of the most important cornerstones of the long standing and very fruitful collaboration of the Austrian Diophantine Number Theory research group and the Number Theory and Cryptography School of Debrecen. However, we do not plan to be complete in any sense but give some interesting data and selected results that we find particularly nice. At the end we focus on two topics in more details, namely a problem that origins from a conjecture of Rényi and Erdős (on the number of terms of the square of a polynomial) and another one that origins from a question of Zelinsky (on the unit sum number problem). This paper evolved from a plenary invited talk that the authors gaveat the Joint Austrian-Hungarian Mathematical Conference 2015, August 25-27, 2015 in Győr (Hungary)

Analysis of linear combination algorithms in cryptography

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 non adjacent forms of the coefficients of the linear combination, the other based on a certain joint sparse form specifically adapted to this problem. Both methods are sped up using the sliding windows approach combined with precomputed lookup tables. We give explicit and asymptotic results for the number of group operations needed assuming uniform distribution of the coefficients. Expected values, variances and a central limit theorem are proved using generating functions. Furthermore, we provide a new algorithm which calculates the digits of an optimal expansion of pairs of integers from left to right. This avoids storing the whole expansion, which is needed with the previously known right to left methods, and allows an online computation

Tilings induced by a class of cubic Rauzy fractals

We study aperiodic and periodic tilings induced by the Rauzy fractal and its subtiles associated with beta-substitutions related to the polynomial x3-ax2-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 topological properties of the Rauzy fractals. As an example, we show that the Rauzy fractals are not homeomorphic to a closed disc as soon as a≤2b-4. The methods presented in this paper may be used to obtain similar results for other classes of substitutions.© 2012 Elsevier B.V. All rights reserved