4 Outer Functions
This chapter currently contains some disconnected fragments of thoughts.
Kahane proves that if there were fewer than \(2n + 1\) rows in the output, then almost no mapping from \(\mathbb {I}^n\) would be injective and thus the KST would not hold, which is a result that is usually credited to Sternfeld a few years later. In addition, Whitney previously proved that a smooth mapping from a space of dimension \(n\) to a space of dimension \(2n + 1\) can be further embedded into a space of dimension \(2n\), which provides some intuition for Vitushkin and Khenkin’s result that the KST cannot hold if all the inner functions were continuously differentiable.
Kaufman proves a mercifully short proof that the KST cannot hold if the genuine Jacobian matrix is of full rank, but this proof presumes on the equicontinuity of the inner functions, which does not hold in our case because \(\psi _{p,q} \notin C^1\left(\mathbb {I}\right)\) if \(q = 0\) or \(q = 2n\) (per Proposition 2.3.3), while \(\psi _{p,q} \in C^2\left(\mathbb {I}\right)\) otherwise (per Proposition 2.4.10). So, the question becomes is equicontinuity actually necessary for Kaufman’s theorem?
The number of distinct outer functions can also be reduced to one by shifting the argument.
\(\chi _q\left(\Psi _q\left(x_0, x_1, \cdots , x_{n - 1}\right)\right) \equiv \chi \left(\Psi _q\left(x_0, x_1, \cdots , x_{n - 1}\right) + q\right)\).
Kahane’s geometric interpretation of the KST goes like this. Let \(\Gamma _p\) denote an “increasing curve” in \(\mathbb {R}^{2n + 1}\) for the equation \(X_q\left(x\right) = \lambda _p \varphi \left(x - \frac{q}{2n}\right)\). Let \(E \equiv \lambda _1 \Gamma + \lambda _2 \Gamma + \dots + \lambda _n \Gamma \), so \(E \subset \mathbb {I}^{2n + 1}\). Then,
The mapping \(\Gamma _1 \times \Gamma _2 \times \dots \times \Gamma _n\) is injective so \(E\) is a distorted cube
\(E\) is an interpolation set in the sense that every continuous function on can be written in the form \(\chi _1\left(X_1\right) + \chi _2\left(X_2\right) + \dots + \chi _{2n}\left(X_{2n}\right)\).
“[A] Helson set in a locally compact abelian group \(G\) is a closed set \(E\) such that the algebra \(A(E)\) of restrictions to \(E\) of functions in \(A(G)\) (Fourier transforms of summable functions on the dual group) coincides with \(C(E)\), the algebra of all continuous functions on \(E\)." (translated from page 91 of Kahane 1978)
“If \(E\) contains the algebraic sum of two infinite sets, \(E\) is not a Helson set.”
See page 91 of Kahane 1978; it has to do with violating a mesh condition.
If \(\gamma \) coincides on any subinterval of \(\mathbb {I}\) with a polynomial of degree less than \(n\), then \(\{ \gamma \left(\Psi _1\right), \gamma \left(\Psi _2\right), \dots , \gamma \left(\Psi _{2n}\right)\} \) is not a Helson set.
Also has do to with the mesh condition.
If \(\gamma \) is an increasing homeomorphism of \(\mathbb {I}\) that does not coincide with a polynomial of degree less than \(n\) on any subinterval of \(\mathbb {I}\), then \(\gamma \left(E\right)\) is quasi-surely a Helson set. Moreover, the condition on \(\gamma \) is necessary and sufficient, and \(\chi _q \in A^+\left(\mathbb {I}\right)\) where \(A^+\left(\mathbb {I}\right)\) is a class of functions \(\chi \left(z\right) = \sum \limits _{j = 0}^\infty \widehat{a}_j e^{2\pi \imath j z}\).
“The proof relies on the following proposition, which is very similar to one given in [Kahane’s 1975 article] p. 233 and can be proven in the same way.”
Let \(B\left(\mathbb {I}\right)\) be a Banach space contained in \(C\left(\mathbb {I}\right)\) and satisfying hypothesis (H): there exists \(c {\gt} 0\) such that, for any \(\delta {\gt} 0\), there exist finite sets \(D_p\subset I\) (for \(p = 1, 2, \ldots , n\)) with at least one point in every subinterval of length \(\delta \), such that the mapping \(D_1\times D_2\times \dots \times D_n \to D_1 + D_2 + \ldots + D_n\) is injective, and for any function \(u\) of modulus 1 on the set \(D_1 + D_2 + \ldots + D_n\), there exists \(\chi \in B\left(\mathbb {I}\right)\), with \(\left|\left|\chi \right|\right|_{B\left(\mathbb {I}\right)} \leq c\) and \(\left|\left|\chi \right|\right|_{C\left(\mathbb {I}\right)} = 1\), where \(\chi = u\) on \(D_1 + D_2 + \ldots + D_n\). Then one can choose \(\chi \in B\left(\mathbb {I}\right)\).
[I]t follows [from Kronecker’s theorem] that any function of modulus 1 on \(\mathbb {I}\) can be written as \(\sum \limits _{j = 1}^\infty a_j e^{\imath j z}\), with \(\sum \limits _{j = 1}^\infty \left|a_j\right| \leq 2\). By applying the proposition, we can therefore take \(\chi \in (A \circ \gamma )\left(\mathbb {I}\right)\) as soon as homeomorphism \(\gamma \) satisfies the condition:
For every \(\delta {\gt} 0\), there exist finite sets \(D_p \subset \mathbb {I}\) (for \(p = 1, 2, \ldots , n\)) at least one point in every subinterval of \(\mathbb {I}\) of length \(\delta \), such that the mapping \(D_1 \times D_2\times \ldots D_n \to D_1 + D_2 + \ldots + D_n\) is injective, and \(\gamma (D_1 + D_2 + \ldots + D_n)\) is a rationally independent set.
Then \(\chi _q\left(\gamma \left(\Psi _q\left(x_0, x_1, \cdots , x_{n - 1}\right)\right)\right) = \sum \limits _{j = 0}^\infty \widehat{a}_j e^{2\pi \imath j \left(\gamma \left(\Psi _q\left(x_0, x_1, \cdots , x_{n - 1}\right)\right) + q\right)} = \sum \limits _{j = 0}^\infty \widehat{a}_j e^{2\pi \imath j \left(\gamma \left(\Psi _q\left(x_0, x_1, \cdots , x_{n - 1}\right)\right)\right)}\) because it is periodic. In practice, we would have to estimate a (truncated) sequence of coefficients to calibrate to a particular target function.