
Binary additive problem with numbers of special type
A. A. Zhukova^{a}, A. V. Shutov^{b} ^{a} Russian Academy of National Economy and Public Administration under the President of the Russian Federation (Vladimir Branch)
^{b} Vladimir State University
Abstract:
In this paper we consider binary additive problem of the form $
n_1 + n_2 = N $ with $ n_1 \in \mathbb {N} (\alpha, I_1)$, $ N_2
\in \mathbb{N} (\beta, I_2) $, where $\mathbb{N} (\alpha, I) = \{n
\in \mathbb{N}: \{n \alpha \} \in I \} $. Main examples of such
sets are the sets of natural numbers with specified ending of
greedy expansion of the number by linear recurrence sequences
associated with Pisot numbers. Besides that, the sets $ \mathbb{N}
(\alpha, I) $ are special cases of quasilattices. Previously
additive problems on the sets of this type are considered only for
the case $ \alpha = \beta $. In this case was obtained asymptotic
formulaes for the number of solutions of the additive problem with
an arbitrary number of terms, and for number of solutions in
analogues of ternary Goldbach problem, HuaLoken problem, Waring
problems, and Lagrange problem about the representation number of
natural numbers as a sum of four squares. Wherein, Gritsenko and
Motkina discovered that in the case of linear problems we have the
following nontrivial effect: apprearence of a rather complicated
function in the main term of the asymptotics for the number of
solutions. For nonlinear problems corrsponding effect is missing
and the form of the main term can be obtained by the density
considerations.
In our problem, we show that the behavior of the main term of the
asymptotic formula for the number of solutions significantly
depends on the arithmetic of $ \alpha $ and $ \beta $. If $ 1 $, $
\alpha $ and $ \beta $ are linearly independent over the ring of
integers $ \mathbb{Z} $, then the main term of the asymptotic has
the "density" form, i.e. it is equal to $  I_1  I_2  N $. In
the case of linear dependence of $ 1 $, $ \alpha $ and $ \beta $
we have the GritsenkoMotkina effect, i.e. the main term is $\rho
(\{N \beta \}) N $, where $ \rho $ is a rather complicated
efficiently computable piecewise linear function of the fractional
part $ \{N \beta \} $. we obtain an algorithm for computation of
the function $ \rho $, and study basic properties of this
function. In particular, we obtain sufficient conditions for its
nonvanishing. Also we give a numerical example of the computation
of this function for some concrete sets $ \mathbb{N} (\alpha, I_1)
$, $ \mathbb {N} (\beta, I_2) $. In the final part of the paper we
discuss some open problems in this area.
Bibliography: 23 titles.
Keywords:
additive problem, uniform distribution.
Full text:
PDF file (3377 kB)
References:
PDF file
HTML file
UDC:
511.34 Received: 02.06.2015
Citation:
A. A. Zhukova, A. V. Shutov, “Binary additive problem with numbers of special type”, Chebyshevskii Sb., 16:3 (2015), 246–275
Citation in format AMSBIB
\Bibitem{ZhuShu15}
\by A.~A.~Zhukova, A.~V.~Shutov
\paper Binary additive problem with numbers of special type
\jour Chebyshevskii Sb.
\yr 2015
\vol 16
\issue 3
\pages 246275
\mathnet{http://mi.mathnet.ru/cheb417}
\elib{http://elibrary.ru/item.asp?id=24398936}
Linking options:
http://mi.mathnet.ru/eng/cheb417 http://mi.mathnet.ru/eng/cheb/v16/i3/p246
Citing articles on Google Scholar:
Russian citations,
English citations
Related articles on Google Scholar:
Russian articles,
English articles

Number of views: 
This page:  139  Full text:  59  References:  24 
