On asymptotic density in \(n\)-dimensions (Q2534219)

The notion of asymptotic density for sets of non-negative integers is generalized to sets of \(n\)-dimensional ``non-negative'' lattice points. Let \(S\) be the set of all \(n\)-tuples of non-negative integers. For \(A\subseteq S\), the Schnirelmann density, \(d(A)\), of \(A\), has been defined to be \(glb_F(A(F)/S(F))\) where \(X(Y)\) is the number of non-zero points in \(X\cap Y\) and \(F\) ranges over all finite sets which are unions of sets of the form \[ L(x_1,\ldots,x_n) = \{(y_1,\ldots,y_n)\mid y_i \le x_i (i=1,2,\ldots,n)\}. \] For a non-negative integer \(N\) let \(J(N)\) be the set of all \((x_1,\ldots,x_n)\) such that \(x_i \le N\) for some \(i\). Then the (lower) asymptotic density of \(A \subset S\) is defined to be \[ \delta(A) = \lim_{n\to\infty} d(A\cup J(N)). \] Various equivalences of this definition are considered which involve some purely combinatorial results estimating the number of lattice points in certain subsets of \(S\). Additive results of the following type are proved: If \((0,\ldots,0) \in A\cap B\) and \(\delta(A) + \delta(B) >1\), then \(\delta(A +B) =1\); \(\delta(A) = \delta(x + A)\) (i.e. \(\delta\) is translation invariant); \(\delta(A + B) \ge \min [1, d_1(A) + \delta(B))\) where \(d_1(A)\) is the generalized Erdős density of \(A\) [see \textit{B. Kvarda} (Garrison), Pac. J. Math. 15, 545--550 (1965; Zbl 0134.27802)]. These results are extended to the infinite dimensional case; \(Q\) is the set of all sequences of non-negative integers all terms except a finite number being zero. For \(A\subset Q\), \(A_n\) is the set of all points \((x_1,\ldots,x_n)\in S\) which can be extended to a sequence in \(A\). Then \(\delta(A)\) is defined to be \(\liminf \delta(A_n)\). Upper asymptotic density \(\bar\delta(A)\) and natural density \(\nu(A)\) \(( = \delta(A)\) if and only if \(\delta(A) = \bar\delta(A))\) are discussed briefly. \(\nu\) is finitely additive and not countably additive. \textit{R. C. Buck}'s ``additivity theorem'' [Am. J. Math. 75, 335--346 (1953; Zbl 0050.05901)] is an open question for this natural density.