Topological dynamics formulation

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Define a topological dynamical system over the rationals to be a pair (X,T), where X is a compact metrisable space, and [math]T = (T_q)_{q \in {\Bbb Q}^+}[/math] is a continuous action of the positive rationals (as a multiplicative group) on X. In other words, for each positive rational q, [math]T_q: X \to X[/math] is a homeomorphism such that [math]T_{qr} = T_q T_r[/math] for all positive rationals q, r. In particular, the [math]T_q[/math] all commute. For any function [math]f: X \to {\Bbb C}[/math], we write [math]T_q f[/math] for [math]f \circ T_q[/math].

The Erdos discrepancy problem is then equivalent to

Conjecture 1. Let (X,T) be a topological dynamical system over the positive rationals, and let [math]f: X \to \{-1,+1\}[/math] be a continuous function. Then the quantity [math] \sum_{i=1}^n T_i f(x)[/math] is unbounded as x ranges over X and n ranges over the natural numbers.

Proof of Conjecture 1 assuming EDP Suppose for contradiction that [math]|\sum_{i=1}^n T_i f(x)| \leq C[/math] for some C and all x, n. Pick a point [math]x_0[/math] in X, and consider the function [math]\tilde f: {\Bbb N} \to \{-1,1\}[/math] defined by

[math]\tilde f(i) := T_i f(x_0).[/math] (1)

Then [math]\tilde f[/math] has discrepancy at most C, contradicting EDP. QED

Proof of EDP assuming Conjecture 1 It suffices to show EDP for the positive rationals. Suppose for contradiction that this failed, then there exists [math]f: {\Bbb Q}^+ \to \{-1,1\}[/math] with discrepancy bounded by some finite C. Let [math]\Omega[/math] be the compact metrisable space [math]\Omega = \{-1,1\}^{{\Bbb Q}^+}[/math] with shift :[math]T_q ( (a_r)_{r \in {\Bbb Q}^+} ) := (a_{qr})_{r \in {\Bbb Q}^+}[/math]; observe that this is a continuous action of the rationals. Let [math]x_0 \in \Omega[/math] be the point

[math]x_0 := (f(r))_{r \in {\Bbb Q}^+}[/math]

and let X be the orbit closure of [math]x_0[/math], i.e. the topological closure of [math]\{ T_q(x_0): q \in {\Bbb Q}^+ \}[/math]. This is a compact metrisable space, and T restricts to a continuous action on this space.

Set [math]\tilde f: X \to \{-1,+1\}[/math] to be the function

[math]\tilde f( (a_r)_{r \in {\Bbb Q}^+} ) := a_1[/math];

observe that this is a continuous function. By Conjecture 1, we can find [math]x = (a_r)_{r \in {\Bbb Q}^+}[/math] and n such that [math]|\sum_{i=1}^n T_i \tilde f(x)| \gt C[/math]. But x can be approximated to arbitrary accuracy by a shift of [math]x_0[/math]. Unpacking all the definitions, we conclude that f has discrepancy greater than C, a contradiction. QED.

We say that a topological system X is minimal if it contains no proper non-empty compact shift-invariant subset. An easy application of Zorn's lemma shows that every topological system contains a minimal system. Thus, to prove Conjecture 1, it suffices to do so for minimal systems.

Given a non-empty open set in a minimal system, one must be able to cover that system by the shifts of the open set, since otherwise the complement of that cover would be a proper compact shift-invariant subset, contradicting minimality. By compactness, this implies that a minimal system can be covered by finitely many translates of the open set.

In terms of sequences, this means that the sequences [math]f: {\Bbb Q}^+ \to \{-1,+1\}[/math] associated to a minimal system (by (1)) have the following almost periodicity property: given any finite set of equations of the form

[math] f(q_1 x) = a_1, \ldots, f(q_k x) = a_k[/math] (*)

for some positive rationals [math]q_1,\ldots,q_k[/math] and [math]a_1,\ldots,a_k\in \{-1,+1\}[/math], the set of solutions x to (*) is either empty or syndetic, which means that there is a finite set of positive rationals [math]r_1,\ldots,r_m[/math] such that for every positive rational x, at least one of [math]xr_1,\ldots,xr_m[/math] solves (*).

The Krylov-Bogolubov theorem asserts that X supports a probability measure that is shift-invariant. The reason for this is that the positive rationals are amenable, and thus admit a Folner sequence F_n. Now start with your favourite probability measure (e.g. a Dirac mass) and average it over the Folner sequences. Then use Prokhorov's theorem to take a weak limit, which will be automatically invariant by construction.

Once we have a shift-invariant measure, ergodic theory comes into play. For instance, the Birkhoff ergodic theorem will assert that for all rationals, and all continuous functions F, the limit [math]\lim_{N\to\infty} \frac{1}{N} \sum_{n=1}^n T_{q^n} F(x)[/math] exists for almost every x in X (with respect to the invariant measure). Because there are only countably many rationals, and the space of continuous functions is separable, we can thus find an x which is generic, in the sense that the above limits exist for all F and all q. In particular, this implies that if EDP fails, we can find a minimal sequence f of bounded discrepancy such that the limit

[math]\lim_{N \to \infty} \frac{1}{N} \sum_{n=1}^n F(f( q^n r_1 ), \ldots, f(q^n r_m))[/math]

exists for all positive rationals [math]q, r_1,\ldots,r_m[/math] and all functions [math]F: \{-1,+1\}^m \to {\Bbb C}[/math].

Note also that if [math]f: X \to \{-1,+1\}[/math] has bounded discrepancy on a measure preserving system, then its mean must be zero, as can be seen by averaging [math]\frac{1}{n} (f(x)+\ldots+f(T_n x))[/math] with respect to x, and then sending n to infinity. Thus, f equals 1 exactly half of the time, and -1 half the time.