83 research outputs found
Good approximation and characterization of subgroups of R/Z
Let α be a real irrational number and A = (x n) be a sequence of positive integers. We call A a characterizing sequence of α or of the group â€Î± mod 1 if lim â„nÎČâ„ = 0 nâA nââ if and only if ÎČ â â€Î± mod 1. In the present paper we prove the existence of such characterizing sequences, also for more general subgroups of â/â€. In the special case â€Î± mod 1 we give explicit construction of a characterizing sequence in terms of the continued fraction expansion of α. Further, we also prove some results concerning the growth and gap properties of such sequences. Finally, we formulate some open problems
On a hybrid fourth moment involving the Riemann zeta-function
We provide explicit ranges for for which the asymptotic formula
\begin{equation*} \int_0^T|\zeta(1/2+it)|^4|\zeta(\sigma+it)|^{2j}dt \;\sim\;
T\sum_{k=0}^4a_{k,j}(\sigma)\log^k T \quad(j\in\mathbb N) \end{equation*} holds
as , when , where is the
Riemann zeta-function. The obtained ranges improve on an earlier result of the
authors [Annales Univ. Sci. Budapest., Sect. Comp. {\bf38}(2012), 233-244]. An
application to a divisor problem is also givenComment: 21 page
Pauli graphs, Riemann hypothesis, Goldbach pairs
Let consider the Pauli group with unitary quantum
generators (shift) and (clock) acting on the vectors of the
-dimensional Hilbert space via and , with
. It has been found that the number of maximal mutually
commuting sets within is controlled by the Dedekind psi
function (with a prime)
\cite{Planat2011} and that there exists a specific inequality , involving the Euler constant , that is only satisfied at specific low dimensions . The set is closely related to
the set of integers that are totally Goldbach, i.e.
that consist of all primes ) is equivalent to Riemann hypothesis.
Introducing the Hardy-Littlewood function (with the twin prime constant),
that is used for estimating the number of
Goldbach pairs, one shows that the new inequality is also equivalent to Riemann hypothesis. In this paper,
these number theoretical properties are discusssed in the context of the qudit
commutation structure.Comment: 11 page
On the critical pair theory in abelian groups : Beyond Chowla's Theorem
We obtain critical pair theorems for subsets S and T of an abelian group such
that |S+T| < |S|+|T|+1. We generalize some results of Chowla, Vosper, Kemperman
and a more recent result due to Rodseth and one of the authors.Comment: Submitted to Combinatorica, 23 pages, revised versio
A quantitative version of the non-abelian idempotent theorem
Suppose that G is a finite group and A is a subset of G such that 1_A has
algebra norm at most M. Then 1_A is a plus/minus sum of at most L cosets of
subgroups of G, and L can be taken to be triply tower in O(M). This is a
quantitative version of the non-abelian idempotent theorem.Comment: 82 pp. Changed the title from `Indicator functions in the
Fourier-Eymard algebra'. Corrected the proof of Lemma 19.1. Expanded the
introduction. Corrected typo
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