It is well-known that the set of functions \(f_ n(x)=nx (mod 1)\), defined in [0,1) and regarded as a subspace of the Tychonoff cube \([0,1]^{[0,1)}\) becomes dense when projected (by restriction) into \([0,1]^ E\) where \(E\subset [0,1)\) and \(E\cup \{1\}\) is independent over the rationals. The authors are concerned with the following problem: a sequence \(n_ 0<n_ 1<..\). in \({\mathbb{Z}}^+\) being given does there exist an uncountable set \(E\subset [0,1)\) such that the set \(\{f_{n_ k}| E\), \(k=0,1,2,...\}\) is dense in \([0,1]^ E\), and in particular does it hold true for \(n_ k=p^ k\) where p is an integer \(\geq 2?\) Instead of \(f_ n\), defined as above, it is more natural to consider the continuous characters of \({\mathbb{T}}\), that is to set \(f_ n(z)=z^ n\) (z\(\in {\mathbb{T}}\), \(n\in {\mathbb{Z}})\). If \(\Phi =\{f_{n_ k}\}\) is any subset of \(\{f_ n\}\) and \(E\subset {\mathbb{T}}\) is such that the set \(\{f_{n_ k}| E\}\) is dense in \({\mathbb{T}}^ E\) then E is called \(\Phi\)-independent. The authors consider this notion in the more general setting of locally compact abelian groups. A set \(E\subset G\) is called independent if for any \(x_ 1,...,x_ n\in E\) and any integers \(k_ 1,...,k_ n\) the equality \(x_ 1^{k_ 1}...x_ n^{k_ n}=1\) implies \(k_ 1=...=k_ n=0\), or else if any function \(E\to {\mathbb{T}}\) extends to a (may be non-continuous) character of G. If \(\Phi\) is a subset of the dual group \(\hat G\) then \(E\subset G\) is called \(\Phi\)- independent if any function \(E\to {\mathbb{T}}\) extends to a character of G contained in the closure of \(\Phi\) in the Bohr compactification bĜ (that is in the pointwise closure). A subset \(\Phi\) is called unbounded if its closure in \(\hat G\) is not compact, so for G compact “\(\Phi\) unbounded” means “\(\Phi\) infinite”. Let us assume that (*) for any integer \(m\neq 0\) and any non-compact \(S\subset \hat G\) the set \(\{\chi^ m:\) \(\chi\in S\}\) is non-compact. Then one has (Theorem 1): for every unbounded \(\Phi\) \(\subset \hat G\) there exists a \(\Phi\)-independent set of cardinality continuum. If, in addition, G is second countable then this set can be chosen to be a dense countable union of Cantor sets.
The proof of this interesting result proceeds in two steps. First one proves (by means of classical Fourier analysis) that, for any \(n\geq 1\), the set \(R_ n\) of all \((x_ 1,...,x_ n)\in G^ n\) such that some f: \(\{\) \(x_ 1,...,x_ n\}\to {\mathbb{T}}\) cannot be extended to a character contained in the pointwise closure of \(\Phi\) is of the first category in \(G^ n\). \(R_ n\) can be viewed as an n-ary relation in G, so one is in a position to apply a theorem of J. Mycielski [Fundam. Math. 55, 139-147 (1964; Zbl 0124.013)] about existence of independent sets of cardinality continuum in topological relation structures.
Another method of obtaining “large” independent sets for \(\Phi\) infinite in compact abelian groups satisfying (*) consists in a transfinite induction. If one assumes the continuum hypothesis that method leads to a \(\Phi\)-independent set E having a non-void intersection with every Borel set which is either of positive Haar measure or of the second Baire category. E is then non-measurable and does not satisfy the Baire condition.