Leírás
A given subset $A$ of natural numbers is said to be complete if every element of $\mathbb{N}$ is the sum of distinct terms taken from $A$. In higher dimension the definition is similar: for any $X \subseteq \mathbb{N}^k$ let $FS(X) :=\{\sum_{i=1}^{\infty} \varepsilon_i x_i\,|\, x_i \in X, \varepsilon_i \in \{0, 1\}, \sum_{i=1}^{\infty} \varepsilon_i <\infty\}$. We say that a set $X$ is complete respect to the region $R\subseteq{N}^k$ if $R\subseteq FS(X)$ holds. A set $X$ is a thin complete set of $R$ if the counting function $X(N) ≤ k log_2 R(N) + t_X$ for some $t_X$ and $FS(X)\supseteq R$. We construct ’thin’ complete set provided the domain $R$ does not contain half-lines parallel to the axis. We investigate the distribution of the subset sum of ’splitable’ sets, the structure of $FS(\{a_m\} × \{b_k\})$ where $\{a_m\}$ is dense and modular version too.
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