Additive transversality of fractal sets in the reals and the integers

Daniel Glasscock, Florian Richter and I have just uploaded to the arXiv our new paper entitled “Additive transversality of fractal sets in the reals and the integers”. In this paper we introduce a new notion of fractal sets of natural numbers in analogy with {\times r}-invariant subsets of {{\mathbb T}={\mathbb R}/{\mathbb Z}}, and obtain the analogues of several results pertaining {\times r}-invariant subsets of {{\mathbb T}={\mathbb R}/{\mathbb Z}}.

The prototypical example of the kind of fractal subsets of {{\mathbb N}_0=\{0,1,2,\dots\}} that we deal with is the integer middle thirds Cantor set consisting of all {n\in{\mathbb N}_0} that when represented in base {3} use only the digits {0} and {2}. Symbolically

\displaystyle C_{\mathbb Z}:=\left\{\sum_{i=0}^na_i3^i:n\in{\mathbb N}_0, (a_0,\dots,a_n)\in\{0,2\}^{n+1}\right\}.

For comparison, the classical middle thirds Cantor set consists of all {x\in[0,1]} that when written in base {3} use only the digits {0} and {2}, and can be described symbolically as

\displaystyle C_{\mathbb R}:=\left\{\sum_{i=1}^na_i3^{-i}:n\in{\mathbb N}_0, (a_1,\dots,a_n)\in\{0,2\}^{n+1}\right\}.

— 1. {\times r}-invariant subsets of {{\mathbb T}}

While there is no universal consensus on what a fractal is, most people would agree that {C_{\mathbb R}} is (perhaps the earliest explicit example of) a fractal set. In particular it has fractional Hausdorff dimension, namely {\dim_H(C_{\mathbb R})=\log 2/\log 3}. It is easy to see that if {x\in C_{\mathbb R}} then {3x\bmod1} is also in {C_{\mathbb R}}; so we say that {C_{\mathbb R}} is a {\times 3}-invariant set. More generally, for any base {r\geq2} we define the class of {\times r}-invariant sets.
Definition 1 Let {r\geq2}. A set {X\subset{\mathbb T}} is said to be {\times r}-invariant if it is closed and {rX=\{rx:x\in X\}\subset X} (where the product {rx} should be understood inside {{\mathbb T}}, or equivalently {b\mod 1}).

The class of {\times r}-invariant subsets of {{\mathbb T}} contains properly the class of restricted digit sets in base {r}, i.e. sets which, like {C_{\mathbb R}}, consist of all numbers {x\in[0,1)} that can be represented in base {r} using only digits from a subset {D\subset\{0,1,\dots,r-1\}} of all digits. Being defined multiplicatively (or, more accurately, in terms of digit expansions) the natural questions to ask about {\times r}-invariant sets are about their relation with the additive structure, and the relation between a {\times r}-invariant set and a {\times s}-invariant set, for distinct {r} and {s}. On the latter, it is important to notice that if {s} is a power of {r}, then any {\times r}-invariant set is also a {\times s}-invariant set. On the other hand, if {r} and {s} are multiplicatively independent, then we expect that a {\times r}-invariant set and a {\times s}-invariant set to share little structure. It turns out that the notion of multiplicative independence we require is very weak (and, for instance, much weaker than being co-prime).

Definition 2 Two numbers {r,s\in{\mathbb N}} are multiplicatively independent if they are not both an integer power of the same integer or, equivalently, if {\log r/\log s\notin{\mathbb Q}}.

Here is a sample of results about {\times r}-invariant sets.

Theorem 3 Let {r,s\in{\mathbb N}} be multiplicatively independent, let {X\subset{\mathbb T}} be a {\times r}-invariant set and let {Y\subset{\mathbb T}} be a {\times s}-invariant set.
  1. If {X\neq{\mathbb T}} then {\dim_H(X)<1}.
  2. (Furstenberg, 1968) If {X=Y}, then {X} is either finite or {X={\mathbb T}}.
  3. (Lindenstrauss-Meiri-Peres, 1999) Letting {X_n=X+\cdots+X\bmod1} be the sum of {n} copies of {X}, {\lim_{n\rightarrow\infty}\dim_H X_n=1}.
  4. (Hochman-Shmerkin, 2012) {\displaystyle \dim_H(X+Y)=\min(\dim_H(X)+\dim_H(Y),1)}
  5. (Shmerkin/ Wu, 2019) {\displaystyle\dim_H(X\cap Y)=\max(\dim_H(X)+\dim_H(Y)-1,0)}.

While I fear that this terse summary of some properties of {\times r}-invariant sets may not be enough to convey how interesting these sets are, and certainly doesn’t highlight the fascinating connections to other areas of mathematics, such as probability and number theory, I will refrain from saying more in order to remain on topic. Suffice it to say that, apart from Furstenberg’s seminal paper essentially starting the whole subject, all the other articles linked above were published in the Annals of Mathematics.

— 2. {\times r}-invariant subsets of {{\mathbb N}_0}

Before talking about fractal subsets of {{\mathbb N}_0} we need a notion of dimension. While fractal subsets of the Euclidean space {{\mathbb R}^n} have the property that zooming in reveals similar features at every scale, fractal subsets of {{\mathbb N}_0} will instead have the property that zooming out will reveal similar features at every scale (this may be thought of as an instance of the duality between compact and discreet sets). The notion of dimension must capture this zooming out idea.

Definition 4 Let {A\subset{\mathbb N}_0}. Its (mass) dimension is

\displaystyle \dim_M(A)=\lim_{N\rightarrow\infty}\frac{\log|A\cap\{1,\dots,N\}|}{\log N}.

Not every subset of {{\mathbb N}_0} has a well defined dimension as the limit used in the definition may not exist. Observe that {{\mathbb N}_0} itself has dimension {1}, as does any set {A\subset{\mathbb N}_0} of positive density. On the other hand, the set of prime numbers, which has zero density, still has dimension {1}. The equation {\dim_M(A)=\gamma} is equivalent to {|A\cap\{1,\dots,N\}|=N^{\gamma+o(1)}}, so the dimension of a set captures its rough growth rate. For instance, letting {Q} be the set of perfect squares we have {\dim_M(Q)=1/2}.

It is not immediately clear how to define an analogous class of {\times r}-invariant subsets of {{\mathbb N}_0}. For instance, the naive definition that stipulates that {rX\subset X} does not lead to an interesting class: if the {X} is the set of all {n\in{\mathbb N}_0} which can be represented in base {r} using only digits from {D\subset\{0,\dots,r-1\}}, then whenever {0\notin D} the containment {rX\subset X} is false. Furthermore, the set {Q} of perfect squares satisfies simultaneously {4Q\subset Q} and {9Q\subset Q}, but {\dim_M(Q)=1/2}.

Instead we need to be a bit more flexible, keeping in mind the duality between compact and discrete sets. For {X\subset{\mathbb T}} we require that {rX\bmod1\subset X}, which can be described as multiplication followed by a natural operation ({\bmod1}) to bring the result back to the compact set. For {X\subset{\mathbb N}_0} we will require that {\lfloor X/r\rfloor\subset X}, which can be described as a division, followed again by a natural operation (the floor function {\lfloor\cdot\rfloor}) to bring the result back to the discrete set. Moreover, for subsets of {{\mathbb T}} we require them to be closed. This is a bit harder to transpose into the discrete world, but we do need an additional condition which is in some sense capturing the same property.

Definition 5 Let {r\geq2} and let {a\in{\mathbb N}_0} have base {r} expansion

\displaystyle a=a_nr^n+a_{n-1}r^{n-1}+\cdots+a_1r+a_0\qquad n\in{\mathbb N}_0,\ a_i\in\{0,\dots,r-1\}\ a_n\neq0.

Define the operations

\displaystyle {\mathfrak L}(a)=a_{n-1}r^{n-1}+\cdots+a_1r+a_0\qquad\qquad {\mathfrak R}(a)=a_nr^{n-1}+a_{n-1}r^{n-2}+\cdots+a_1.

A set {A\subset{\mathbb N}_0} is said to be {\times r}-invariant if both {{\mathfrak L}(A)\subset A} and {{\mathfrak R}(A)\subset A}.

Note that {{\mathfrak L}} deletes the Left-most (or, Leading) digit, while {{\mathfrak R}} deletes the Right most digit (or the Remainder upon division by {r}).

Any base {r} restricted digit Cantor set in the integers is a {\times r}-invariant set. In particular, {C_{\mathbb Z}} is a {\times 3}-invariant set and, unsurprisingly, its mass dimension is {\dim_M(C_{\mathbb Z})=\log 2/\log 3}.

Using this notion {\times r}-invariant sets in {{\mathbb N}_0}, we obtain analogues of most items in Theorem 3.

Theorem 6 Let {r,s\in{\mathbb N}} be multiplicatively independent, let {A\subset{\mathbb N}_0} be a {\times r}-invariant set and let {B\subset{\mathbb N}_0} be a {\times s}-invariant set.
  1. {\dim_M(A)} exists.
  2. If {A\neq{\mathbb N}_0} then {\dim_M(A)<1}.
  3. If {A=B}, then {A} is either finite or {A={\mathbb N}_0}.
  4. Letting {A_n=A+\cdots+A\bmod1} be the sum of {n} copies of {A}, {\lim_{n\rightarrow\infty}\dim_M A_n=1}.
  5. {\displaystyle \dim_M(A+B)=\min(\dim_M(A)+\dim_M(B),1)}

Both Theorems 3 and 6 are formulated only for a special representative case of the full strength of those theorems.

Regarding item 3. in Theorem 3, we believe that the analogue for {\times r}-invariant subsets of {{\mathbb N}_0} also holds and hope to have a proof for it soon, but we are not yet ready to announce a result.

— 3. Proof outline and a strengthening of the Hochman-Shmerkin theorem —

When we started the project, my hope was that after figuring out what the definition for a {\times r}-invariant sets would be, we would be able to use some sort of correspondence to directly derive Theorem 6 from Theorem 3. This turned out to be more complicated than anticipated: for parts 6. and 6. we found direct proofs, which don’t rely on the counterparts on {{\mathbb T}}. In part 6. things worked as expected: we were able to derive it from the analogous part 3. from Theorem 3 by a sort of correspondence.

However in order to prove part 6. of Theorem 6 with a similar correspondence, we needed a stronger version of part 3. from Theorem 3. This stronger version is a bit technical to state formally here (it’s Theorem A in the paper) and, while it is stronger in a precise quantitative sense than part 3. of Theorem 3, the full strength of the results of Hochman and Shmerkin are significantly more general in a different sense. Nevertheless, it does not seem possible to obtain our quantitative strengthening from the results of Hochman and Shmerkin alone, so we ended up giving a different and direct proof of this theorem, which has the advantage of being mostly elementary and avoiding most of the heavy machinery typically necessary in this area.

Finally, the integer analogue of part 3. of Theorem 3 seems to require a similarly stronger version of its counterpart in {{\mathbb T}}.

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