Binary additive problem with numbers of special type

A. A. Zhukova ; A. V. Shutov

Čebyševskij sbornik, Tome 16 (2015) no. 3, pp. 246-275.

Résumé

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, Hua-Loken 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 Gritsenko-Motkina 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 non-vanishing. 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.

Détail

Keywords: additive problem, uniform distribution.

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