Normality of numbers generated by the values of entire functions

AbstractWe show that the number generated by the q-ary integer part of an entire function of logarithmic order, where the function is evaluated over the natural numbers and the primes, respectively, is normal in base q. This is an extension of related results for polynomials over the real numbers established by Nakai and Shiokawa

A Meinardus theorem with multiple singularities

Meinardus proved a general theorem about the asymptotics of the number of weighted partitions, when the Dirichlet generating function for weights has a single pole on the positive real axis. Continuing \cite{GSE}, we derive asymptotics for the numbers of three basic types of decomposable combinatorial structures (or, equivalently, ideal gas models in statistical mechanics) of size $n$, when their Dirichlet generating functions have multiple simple poles on the positive real axis. Examples to which our theorem applies include ones related to vector partitions and quantum field theory. Our asymptotic formula for the number of weighted partitions disproves the belief accepted in the physics literature that the main term in the asymptotics is determined by the rightmost pole.Comment: 26 pages. This version incorporates the following two changes implied by referee's remarks: (i) We made changes in the proof of Proposition 1; (ii) We provided an explanation to the argument for the local limit theorem. The paper is tentatively accepted by "Communications in Mathematical Physics" journa

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

Active Faulting in Lake Constance (Austria, Germany, Switzerland) Unraveled by Multi-Vintage Reflection Seismic Data

Probabilistic seismic hazard assessments are primarily based on instrumentally recorded and historically documented earthquakes. For the northern part of the European Alpine Arc, slow crustal deformation results in low earthquake recurrence rates and brings up the necessity to extend our perspective beyond the existing earthquake catalog. The overdeepened basin of Lake Constance (Austria, Germany, and Switzerland), located within the North-Alpine Molasse Basin, is investigated as an ideal (neo-) tectonic archive. The lake is surrounded by major tectonic structures and constrained via the North Alpine Front in the South, the Jura fold-and-thrust belt in the West, and the Hegau-Lake Constance Graben System in the North. Several fault zones reach Lake Constance such as the St. Gallen Fault Zone, a reactivated basement-rooted normal fault, active during several phases from the Permo-Carboniferous to the Mesozoic. To extend the catalog of potentially active fault zones, we compiled an extensive 445 km of multi-channel reflection seismic data in 2017, complementing a moderate-size GI-airgun survey from 2016. The two datasets reveal the complete overdeepened Quaternary trough and its sedimentary infill and the upper part of the Miocene Molasse bedrock. They additionally complement existing seismic vintages that investigated the mass-transport deposit chronology and Mesozoic fault structures. The compilation of 2D seismic data allowed investigating the seismic stratigraphy of the Quaternary infill and its underlying bedrock of Lake Constance, shaped by multiple glaciations. The 2D seismic sections revealed 154 fault indications in the Obersee Basin and 39 fault indications in the Untersee Basin. Their interpretative linkage results in 23 and five major fault planes, respectively. One of the major fault planes, traceable to Cenozoic bedrock, is associated with a prominent offset of the lake bottom on the multibeam bathymetric map. Across this area, high-resolution single channel data was acquired and a transect of five short cores was retrieved displaying significant sediment thickness changes across the seismically mapped fault trace with a surface-rupture related turbidite, all indicating repeated activity of a likely seismogenic strike-slip fault with a normal faulting component. We interpret this fault as northward continuation of the St. Gallen Fault Zone, previously described onshore on 3D seismic data

Microstructural characterization of natural fractures and faults in the Opalinus Clay: insights from a deep drilling campaign across central northern Switzerland

Abstract The Middle-Jurassic Opalinus Clay is the foreseen host rock for radioactive waste disposal in central northern Switzerland. An extensive drilling campaign aiming to characterize the argillaceous formation resulted in a comprehensive drill core data set. The rheologically weak Opalinus Clay is only mildly deformed compared to the over- and underlying rock units but shows a variety of natural fractures. While these structures are hydraulically indistinguishable from macroscopically non-deformed Opalinus Clay today, their analysis allows for a better understanding of the deformation behaviour in the geological past. Here, we present an overview of the different fracture and fault types recorded in the Opalinus Clay and a detailed microstructural characterization of veins—natural dilational fractures healed by secondary calcite and celestite mineralizations. Macroscopic drill core analysis revealed five different natural fracture types that encompass tension gashes of various orientations with respect to bedding and small-scale faults with displacements typically not exceeding the drill core diameter. The occurrence of different fault types generally fits well with the local tectonic setting of the different drilling sites and with respect to the neighbouring regional fault zones. The microstructural investigations of the various vein types revealed their often polyphase character. Fibrous bedding-parallel veins of presumable early age were found to be overprinted by secondary slickenfibres. The polyphase nature of fibrous bedding parallel veins and slickenfibres is supported by differing elemental compositions, pointing towards repeated fracturing and mineralization events. Direct dating of vein calcites with U–Pb was unsuccessful. Nevertheless, age constraints can be inferred from structural orientations and fault slip kinematics. Accordingly, some of the veins already formed during sediment compaction in Mesozoic times, others possibly relate to Early Cenozoic foreland uplift. The youngest veins are most likely related to Late Cenozoic regional tectonic events, such as the Jura fold-and-thrust belt to the south and the Hegau-Lake Constance Graben to the northeast of the study area. During these latest tectonic events, previously formed veins acted as rheologically stiff discontinuities in the otherwise comparably weak Opalinus Clay along which deformation of the rock formation was re-localized

Linking the northern Alps with their foreland: The latest exhumation history resolved by low-temperature thermochronology

The evolution of the Central Alpine deformation front (Subalpine Molasse) and its undeformed foreland is recently debated because of their role for deciphering the late orogenic evolution of the Alps. Its latest exhumation history is poorly understood due to the lack of late Miocene to Pliocene sediments. We constrain the late Miocene to Pliocene history of this transitional zone with apatite fission track and (U-Th)/He data. We used laser ablation inductively coupled mass spectrometry for apatite fission track dating and compare this method with previously published and unpublished external detector method fission track data. Two investigated sections across tectonic slices show that the Subalpine Molasse was tectonically active after the onset of folding of the Jura Mountains. This is much younger than hitherto assumed. Thrusting occurred at 10, 8, 6–5 Ma and potentially thereafter. This is contemporaneous with reported exhumation of the External Crystalline Massifs in the central Alps. The Jura Mountains and the Subalpine Molasse used the same detachments as the External Crystalline Massifs and are therefore kinematically coupled. Estimates on the amount of shortening and thrust displacement corroborate this idea. We argue that the tectonic signal is related to active shortening during the late stage of orogenesis

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)

The General Sieve Kernel and New Records in Lattice Reduction

textabstractWe propose the General Sieve Kernel (G6K, pronounced /Ze.si.ka/), an abstract stateful machine supporting a wide variety of lattice reduction strategies based on sieving algorithms. Using the basic instruction set of this abstract stateful machine, we first give concise formulations of previous sieving strategies from the literature and then propose new ones. We then also give a light variant of BKZ exploiting the features of our abstract stateful machine. This encapsulates several recent suggestions (Ducas at Eurocrypt 2018; Laarhoven and Mariano at PQCrypto 2018) to move beyond treating sieving as a blackbox SVP oracle and to utilise strong lattice reduction as preprocessing for sieving. Furthermore, we propose new tricks to minimise the sieving computation required for a given reduction quality with mechanisms such as recycling vectors between sieves, on-the-fly lifting and flexible insertions akin to Deep LLL and recent variants of Random Sampling Reduction. Moreover, we provide a highly optimised, multi-threaded and tweakable implementation of this machine which we make open-source. We then illustrate the performance of this implementation of our sieving strategies by applying G6K to various lattice challenges. In particular, our approach allows us to solve previously unsolved instances of the Darmstadt SVP (151, 153, 155) and LWE (e.g. (75, 0.005)) challenges. Our solution for the SVP-151 challenge was found 400 times faster than the time reported for the SVP-150 challenge, the previous record. For exact SVP, we observe a performance crossover between G6K and FPLLL’s state of the art implementation of enumeration at dimension 70