Let m, n be positive integers and let : Z n ร R be a non-negative function. Let W(m, n; ) be the set { X # R mn : " : n j=1
x ij q j " < (q), 1 i m, for infinitely many q # Z n = .
The Hausdorff dimension of W(m, n; ) is obtained for arbitrary non-negative functions , with no monotonicity assumptions.
1998 Academic Press
Let m, n be positive integers and let : Z n ร R be a non-negative function. Let W(m, n; ) be the set of points X=(x 11 , ..., x 1n , ..., x m 1 , ..., x mn ) # R mn for which the inequalities " : n j=1
x ij q j " < (q), 1 i m,
( 1 ) hold for infinitely many integer vectors q=(q 1 , ..., q n ) # Z n (where, for any z # R, &z& denotes the distance from z to the nearest integer). In the special case that has the form { (q) :=|q| &{ , q{0, where |q|=max[|q j |: j=1, ..., n] and { 0, the set W(m, n; { ) has been studied by many authors. In particular, for {>nรm its Hausdorff dimension is m(n&1)+(m+n)ร(1+{), see [2] (see also the references in [3] and [7] for papers dealing with special cases of m and n and for other results on W(m, n; { )).
In [5] Eggleston generalized the set W(m, 1; { ) by requiring that the integers q in the definition belong to a given sequence of integers. To be precise, for Q/Z n , { 0, let Q, { (q) := { |q| &{ , 0, when q # Q, when q ร Q Article No. NT982253