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Register a new userM{o}bius disjointness for a class of exponential functions
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A vast class of exponential functions is showed to be deterministic. This class includes functions whose exponents are polynomial-like or piece-wise close to polynomials after differentiation. Many of these functions are indeed disjoint from the Mobius function. As a consequence, we show that Sarnaks Disjointness Conjecture for the Mobius function (from deterministic sequences) is equivalent to the disjointness in average over short intervals
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We provide a criterion for a point satisfying the required disjointness condition in Sarnaks Mobius Disjointness Conjecture. As a direct application, we have that the conjecture holds for any topological model of an ergodic system with discrete spectrum.
A sequential dynamical system (SDS) consists of a graph, a set of local functions and an update schedule. A linear sequential dynamical system is an SDS whose local functions are linear. In this paper, we derive an explicit closed formula for any linear SDS as a synchronous dynamical system. We also show constructively, that any synchronous linear system can be expressed as a linear SDS, i.e. it can be written as a product of linear local functions. Furthermore, we study the connection between linear SDS and the incidence algebras of partially ordered sets (posets). Specifically, we show that the M{o}bius function of any poset can be computed via an SDS, whose graph is induced by the Hasse diagram of the poset. Finally, we prove a cut theorem for the M{o}bius functions of posets with respect to certain chain decompositions.
We obtain an estimate for the average value of the product of the Mobius function and any polynomial phase over short intervals and arithmetic progressions simultaneously. As a consequence, we prove that the product of M{o}bius and any polynomial phase is disjoint from arithmetic functions realized in certain rigid dynamical systems, such as any finite products of translations of Mobius squared.
We establish cancellation in short sums of certain special trace functions over $mathbb{F}_q[u]$ below the P{o}lya-Vinogradov range, with savings approaching square-root cancellation as $q$ grows. This is used to resolve the $mathbb{F}_q[u]$-analog of Chowlas conjecture on cancellation in M{o}bius sums over polynomial sequences, and of the Bateman-Horn conjecture in degree $2$, for some values of $q$. A final application is to sums of trace functions over primes in $mathbb{F}_q[u]$.
We study the topological properties of Peierls transitions in a monovalent M{o}bius ladder. Along the transverse and longitudinal directions of the ladder, there exist plenty Peierls phases corresponding to various dimerization patterns. Resulted from a special modulation, namely, staggered modulation along the longitudinal direction, the ladder system in the insulator phase behaves as a ``topological insulator, which possesses charged solitons as the gapless edge states existing in the gap. Such solitary states promise the dispersionless propagation along the longitudinal direction of the ladder system. Intrinsically, these non-trivial edges states originates from the Peierls phases boundary, which arises from the non-trivial $mathbb{Z}^{2}$ topological configuration.
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