Two dusts that add up to a line

Remove the middle third of an interval, then the middle third of each of the two pieces left, and so on for ever. What survives is the middle-thirds Cantor set: uncountably many points, total length nought, and — as a dimension that is not a whole number measured — dimension $\log 2/\log 3 \approx 0.631$, more than a finite set of points and less than an interval. It is the standard example of a dust.

Now add it to itself. Take every pair of points $x$ and $y$ in the set and form $x + y$. The sums of a set of length nought might be expected to have length nought too, or at most to be another dust of somewhat larger dimension. They are not.

The sums fill the whole interval from 0 to 2. At every stage of the construction the pieces’ sums overlap to leave no gap, and so the limit leaves none either. A set with no length, added to itself, is a line.

The digits say why #

There is a one-line explanation in base 3. A number lies in the Cantor set exactly when it has a base-3 expansion using only the digits 0 and 2 — removing the middle third removes the numbers whose first digit must be 1, and so on at every stage. Halve such a number and its digits are 0s and 1s. So $\tfrac12 C$ is the set of base-3 expansions with digits 0 and 1, and

$$\tfrac12 C + \tfrac12 C$$

is the set of expansions whose digits are sums of two such digits: 0, 1 or 2, freely. That is every number from 0 to 1. Doubling gives $C + C = [0, 2]$. The same argument with a minus sign shows that $C - C = [-1, 1]$: every number between $-1$ and 1 is a difference of two points of the Cantor set. Hugo Steinhaus, who noticed it in 1917, used the fact to show that a set of measure nought can still have every distance appear between two of its points.

Steinhaus’s interest was in a theorem of his own. For any set of positive length, the differences $A - A$ contain a whole interval around nought — a set with length cannot avoid any short distance. The Cantor set shows that the converse fails: length is sufficient for the differences to fill an interval, and not necessary. Length and size of sums are different kinds of bigness, and a set with no interval in it and length to spare makes the opposite point, a set with positive length and no interval at all. Between them the two examples separate three properties that are easy to run together: having length, containing an interval, and having sums that contain one.

The digits explain this particular set, and they hide the general mechanism, which is geometric and applies to Cantor sets with no digit structure at all.

Thinner dusts stay dust #

The middle-thirds set keeps two pieces of ratio $\tfrac13$ at every stage. Keep two end pieces of ratio $\lambda$ instead, for any $\lambda$ below a half, and the result is a Cantor set $C_\lambda$ of dimension $\log 2/\log(1/\lambda)$: thinner for smaller $\lambda$, fatter for larger.

The self-sum $C_\lambda + C_\lambda$ is built from three copies of itself, each shrunk by $\lambda$ — the sums of two left pieces, of a left and a right, and of two right pieces — so its $n$-th stage has $3^n$ pieces of length $2\lambda^n$. When $\lambda < \tfrac13$ the three copies cannot cover the interval, the total length is at most $2(3\lambda)^n$, and it falls to nought: at $\lambda = 0.3$ the tenth stage has length 0.70, at $0.26$ it has 0.17, and both keep falling. At $\lambda = \tfrac13$ the three copies just touch, and at $0.36$ they overlap; in both the sum is the whole interval at every stage.

Why a third is easy to see. The three copies are the interval $[0, 2]$ shrunk by $\lambda$ and placed at $0$, $1 - \lambda$ and $2(1 - \lambda)$, so each has length $2\lambda$ and consecutive copies start $1 - \lambda$ apart. They touch exactly when $2\lambda \ge 1 - \lambda$, that is when $\lambda \ge \tfrac13$, and once they touch at the first stage they touch at every stage, because each stage is the first one shrunk. Below a third there is a gap between the copies at the first stage, and gaps within gaps at every later one.

So the switch is sharp. Below a third, the sum of a Cantor set with itself is again a set of length nought; from a third up, it is an interval. There is no middle ground of sums with some length and some gaps — and the threshold has nothing to do with the dimensions adding to one, which happens at $\lambda = \tfrac14$.

Thickness: a bridge against a gap #

The general reason was found by Sheldon Newhouse in 1970, in the middle of a problem about the chaotic dynamics of maps of the plane, and it is a single number attached to a Cantor set.

Every gap of a Cantor set is flanked by bridges — the stretches of the set’s hull between this gap and the next larger one on each side. The thickness is the least ratio, over all gaps, of a neighbouring bridge to the gap itself. For $C_\lambda$ every gap looks the same, with bridge $\lambda$ and gap $1 - 2\lambda$ at the first stage and the same ratio at every later one, so the thickness is $\lambda/(1 - 2\lambda)$: exactly 1 for the middle-thirds set, a half for $\lambda = \tfrac14$, 2 for $\lambda = 0.4$.

Newhouse’s gap lemma says that two Cantor sets whose thicknesses multiply to at least 1 cannot be interleaved without touching. If neither lies wholly inside a gap of the other, they share a point. The proof is a chase: if they did not meet, some gap of one would have to contain a bridge of the other, and the thickness inequality says every bridge is too long to fit in any gap it could be facing. For sums the lemma applies directly, because $t \in A + B$ exactly when $A$ and $t - B$ meet.

The lemma, tried on many pairs #

The figure tries the lemma on thirty-six pairs. For each pair, slide one set across the other through 801 shifts and ask whether, at the eighth stage, the pieces of the two still overlap anywhere; if they do not, the limiting sets cannot meet either. To the right of the line, where the thicknesses multiply to 1 or more, no shift ever separates them — as the gap lemma promises. To the left the sets slip past each other for a share of the shifts that grows as the thickness falls: a few per cent just left of the line, and 99% when both sets keep pieces of ratio 0.2.

The pair on the line itself is worth singling out. A set keeping pieces of ratio 0.4 has thickness 2 and dimension 0.76; a set keeping pieces of ratio a quarter has thickness a half and dimension exactly a half. Their thicknesses multiply to exactly 1, and no shift separates them: the very thin quarter-set is caught by the very thick 0.4-set every time, because each gap of the thin set is bridged by a stretch of the thick one long enough to reach across. Thickness, unlike dimension, can be traded between the two sets.

The line is sharp in the figure but the lemma is only a sufficient condition. Two thin Cantor sets can still meet for most shifts, and the points just left of the line show it: their thicknesses fall short and their separating shifts are a small minority. Thickness guarantees intersection; it does not measure it.

Where a dust meets its own translate #

The middle-thirds set has thickness exactly 1, so it meets every translate of itself that it is interleaved with — every shift $t$ between $-1$ and 1, the set $C - C$. But how much it meets varies enormously. At $t = 0$ all 512 pieces at the ninth stage meet themselves; at $t = \tfrac23$ and at $t = \tfrac29$, half of them do, because shifting by those amounts lines up one half of the set with the other; at the median shift on the grid only a few dozen do. The intersection $C \cap (C + t)$ is itself a Cantor set whose size is decided by the base-3 digits of $t$: each digit 0 in $t$’s expansion keeps both halves at that stage, and each digit $\pm 2$ keeps one. The figure’s spiky skyline is that digit count, drawn.

The same structure is behind the staircase that is flat almost everywhere: the Cantor set carries a natural distribution of mass, the Cantor function is its running total, and the intersections with translates are where the distribution’s self-similarity becomes a statement about arithmetic.

A sum is a shadow #

There is a geometric way to see every sum in this essay. The product $A \times B$ is a dust in the plane — for two middle-thirds sets it is the four-corner set of squares within squares — and the sum $A + B$ is the shadow of that dust on a line, cast by light falling at forty-five degrees: the point $(x, y)$ lands at $x + y$. Shadows cast at other angles give the weighted sums $A + cB$. So asking whether $C + C$ is an interval is asking whether the four-corner dust of the middle-thirds set casts a solid shadow in one particular direction.

That is the question a dust that almost every line misses asked about a thinner dust, the one that keeps four corner squares of side a quarter. There the dimension of the product is exactly 1, and its shadows in almost every direction have length nought; in a few special directions, set by the base-four digits, they keep a positive length. Here the middle-thirds product has dimension $2\log 2/\log 3 \approx 1.26$, more than 1, and its shadow at forty-five degrees is a full interval. Thickness is a way of guaranteeing a solid shadow; dimension above one is a way of making one likely; the special directions where shadows misbehave are where arithmetic intervenes.

When dimensions add, and when they do not #

For sets with no special relation, the dimension of a sum is expected to be the sum of the dimensions, capped at 1. Jacob Palis conjectured in the 1980s that for typical pairs of Cantor sets there are only two outcomes: if the dimensions add to less than 1 the sum has length nought, and if to more than 1 it contains an interval. Yuval Peres and Pablo Shmerkin proved in 2009 that for self-similar sets like $C_a$ and $C_b$ with ratios whose logarithms are not in a rational proportion, the dimension of the sum is exactly $\min(1, \dim C_a + \dim C_b)$; Carlos Gustavo Moreira and Jean-Christophe Yoccoz proved the interval half of Palis’s conjecture for typical sets arising in dynamics.

A set added to itself is the opposite of typical, and the figure shows what that costs. The box-counting estimate at the ninth stage tracks $\log 3/\log(1/\lambda)$ — the dimension of a set made of three copies shrunk by $\lambda$ — to within a few thousandths at every $\lambda$, and falls well below the dashed curve of twice the dimension of $C_\lambda$ whenever $\lambda < \tfrac13$. At $\lambda = 0.26$ the two summands have dimensions adding to 1.03, and their sum has dimension 0.82 and length nought. The shortfall is arithmetic: the sum of a left piece and a right piece lands in the same place as the sum of a right piece and a left, so four combinations make only three distinct copies. It is the same coincidence that makes two sign patterns land together in sums of $\pm \lambda^n$, and the study of a coin in front of every power is the study of exactly these overlaps.

The same game with whole numbers #

Sums of thin sets that fill everything have a counterpart among whole numbers, where they are a central topic of additive combinatorics. The numbers below $3^n$ whose base-3 digits are only 0 and 1 form a set of $2^n$ elements, a vanishing share of the $3^n$ numbers, and yet every number below $3^n$ is a sum of two of them, since digits 0, 1, 2 are sums of two digits from 0 and 1 with no carrying. A set of size $N^{0.63}$ whose sums reach all of a range of size $N$ is the whole-number shadow of the Cantor set’s sum.

Whole-number sums also have a floor that real sums do not. Modulo a prime, two sets added cannot be small: the Cauchy–Davenport theorem guarantees that the sums of sets of sizes $a$ and $b$ number at least $a + b - 1$, unless they fill everything. Among the real numbers no such floor exists — two Cantor sets of dimension a quarter add to a set of dimension at most a half and length nought — and the interest shifts from how many sums there are to whether they leave gaps, which is exactly the question thickness answers.

What the stages cannot show #

Every figure computes a finite stage — six, eight, nine or ten levels of removal — and the theorems are about the limit. Some statements pass cleanly from stage to limit: if the stage-$n$ pieces of two sets do not overlap, the sets do not meet; if the stage-$n$ sums cover an interval at every $n$, so does the limit. Others do not: a stage at which two sets still overlap says nothing about whether they meet in the end, which is why the gap-lemma figure counts only separations and calls them a lower bound. The dimension estimates use one stage and one box size, and agree with the formula because these sets are exactly self-similar; for a Cantor set without that regularity a single stage would mislead.

The figures also say nothing about the dynamical setting where thickness first mattered. Newhouse used the gap lemma to show that a map of the plane can have infinitely many stable cycles at once, near a tangency of its stable and unstable curves, whenever two Cantor sets attached to the tangency are thick enough to be forced to meet — a phenomenon that persists under perturbation and is one of the reasons the study of chaotic maps in two dimensions is so much harder than in one.

Still open: the middle ground #

For sums of Cantor sets whose dimensions add to more than 1 but whose thicknesses multiply to less than 1, when does the sum contain an interval? Palis’s conjecture says the answer is “almost always”, and it is proved for typical sets in several precise senses, but for specific sets — the self-similar $C_a + C_b$ with ratios in a rational proportion, or sets defined by explicit dynamics — it can fail in either direction, and there is no criterion in terms of simple invariants like dimension and thickness that decides it. The question whether $C_a + C_b$ has positive length but no interval for some specific pair, and how to tell, connects to the open problems about overlapping self-similar sets whose dimension computations are still being completed.

A threshold of one #

The middle-thirds Cantor set has no length, and every number from 0 to 2 is a sum of two of its points. The digits explain it — sums of 0s and 2s produce every base-3 expansion — and Newhouse’s thickness explains why it is not an accident: a bridge-to-gap ratio of 1, and two Cantor sets whose thicknesses multiply to at least 1 cannot pass through each other without meeting. Thinner self-similar dusts summed with themselves fall back to dust, with length shrinking like $(3\lambda)^n$ and dimension $\log 3/\log(1/\lambda)$, short of twice their own because a set added to itself counts each mixed sum twice.