In this paper we consider a problem of distribution of fractional parts of the sequence obtained as multiples of some irrational number with bounded partial quotients of its continued fraction expansion. Local discrepancies are the remainder terms of asymptotic formulas for the number of points of the sequence lying in given intervals. Earlier only intervals with bounded and logarithmic local discrepancies were known. We prove that there exists an infinite set of intervals with arbitrary small growth rate of local discrepancies. The proof is based on the connection of considered problem with some of those from Diophantine approximations.

Local discrepancies in the problem of fractional parts distribution of a linear function

The purpose of this paper is to discuss the relationship between prime numbers and sums of Fibonacci numbers. One of our main results says that for every sufficiently large integer $k$ there exists a prime number that can be represented as the sum of k different and non-consecutive Fibonacci numbers. This property is closely related to, and based on, a prime number theorem for certain morphic sequences. In our case, these morphic sequences are based on the Zeckendorf sum-of-digits function $z$, which returns the number of summands in the representation of a nonnegative integer as sum of non-consecutive Fibonacci numbers. 
The proof of such a prime number theorem, combined with a corresponding local result, constitutes the central contribution of this paper, from which the result stated in the beginning follows. 
These kinds of problems have been discussed intensively in the context of the base-$q$ expansion of integers. The Gelfond problems (1968/1969), and the Sarnak conjecture, were the driving forces of this development. 
Mauduit and Rivat resolved the question on the sum of digits of prime numbers (2010) and the sum of digits of squares (2009), thus leaving open only part of the third Gelfond problem. Later the second author (2017) proved Sarnak's conjecture for the class of automatic sequences, which are based on the $q$-ary expansion of integers, and which generalize the sum-of-digits function in base $q$ considerably. 
In the present paper, we use Gowers norms, and extend Mauduit and Rivat's method significantly. Employing Gowers norms in our proof allows us to prove that $\exp(2\pi i \alpha z(n))$ has level of distribution 1. This latter result forms an essential part of our treatment of the occurring sums of type I and II. An analogous result for the Thue--Morse sequence has been obtained by the third author (2020).

Primes as sums of Fibonacci numbers

Nonautonomous bounded remainder sets are sequences of sets that admit a uniform estimation of the remainder term in the distribution of fractional parts of a linear function. In this paper, we give a complete description of nonautonomous bounded remainder sets in the case of periodic sequences. The result is also extended to certain classes of quasiperiodic sequences of sets. Our proofs are based on obtaining explicit formulas for the remainder term by using sums of fractional parts. This method is effective, i.e., it allows us to explicitly estimate the remainder term.

Nonautonomous Bounded Remainder Sets

We consider the long-time behavior of the trajectories of the discontinuous analog of the standard Chirikov map. We prove that for some values of parameters all the trajectories remain bounded for all time. For other set of parameters we provide an estimate for the escape rate for the trajectories and present a numerically supported conjecture for the actual escape rate.

On a Very Steep Version of the Standard Map

/pdf/a-curiosity-about-1-e-1-2-e-1-ne-5atkxw1sys.pdf

A curiosity About (−1)[ e ] +(−1)[2 e ] + ··· +(−1)[ Ne ]

pdf/on-the-local-discrepancy-of-na-sequences-52qtj9qvja.pdf

On the local discrepancy of (nα)-sequences

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On the local discrepancy of (nα)-sequences