A note on Diophantine approximation

We prove the existence of a dense subset Δ of [ 0 , 4 ] such that for all α ∈ Δ there exists a subgroup X α of infinite rank of ℤ [ z ] such that X α is a discrete subgroup of C [ 0 , β ] for all β ≥ α but it is not a discrete subgroup of C [ 0 , β ] for any β ∈ ( 0 , α ) .Given a set of nonnegative real numbers Λ = { λ i } i = 0 ∞ , a Λ -polynomial (or Muntz polynomial) is a function of the form p ( x ) = ∑ i = 0 n a i z λ i ( n ∈ ℕ ). We denote by Π ( Λ ) the space of Λ -polynomials and by Π ℤ ( Λ ) : = { p ( x ) = ∑ i = 0 n a i z λ i ∈ Π ( λ ) : a i ∈ ℤ  for all  i ≥ 0 } the set of integral Λ -polynomials. Clearly, the sets Π ℤ ( Λ ) are subgroups of infinite rank of ℤ [ x ] whenever Λ ⊂ ℕ , # Λ = ∞ (by infinite rank, we mean that the real vector space spanned by X does not have finite dimension. In all what follows we are uniquely interested in groups of infinite rank). Now, it is well known that the problem of approximation of functions on intervals [ a , b ] by polynomials with integral coefficients is solvable only for intervals [ a , b ] of length smaller than four and functions f which are interpolable by polynomials of ℤ [ x ] on a certain set (which we call the algebraic kernel of the interval [ a , b ] )   ( a , b ) . Concretely, it is well known that ℤ [ x ] is a discrete subgroup of C [ a , b ] whenever b − a ≥ 4 and 4 is the smallest number with this property (for these and other interesting results about approximation by polynomials with integral coefficients, see [1,3] and the references therein. See also the other references at the end of this note). This motivates the following concept.