A nearly full interval can avoid every scaled geometric sequence

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Family 084 · Real and complex analysis
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-14

1. The question

Take a measurable subset of the real line with positive length. Must it contain a translated and rescaled copy of any pattern you choose? For a finite pattern, the answer is yes. A sufficiently small version can fit inside even a rather irregular set of positive measure.

An infinite pattern is different. Erdős conjectured that every infinite set can be defeated: some positive-measure set contains none of its translated and nontrivially rescaled copies. One particularly resistant example was the geometric sequence

$$\frac12,\frac14,\frac18,\ldots.$$

Its points crowd together exponentially near zero. Removing a small amount of material while hitting every possible translated and rescaled copy is consequently difficult.

The two manuscripts in this family claim first the dyadic case, with ratio $1/2$, and then every fixed geometric ratio. Their avoiding sets are compact and can occupy almost all of the unit interval. The later claim covers each ratio separately; it does not say that a single set simultaneously avoids every geometric progression.

2. Background and essential definitions

Lebesgue measure, written $m(E)$, is the usual notion of length for measurable subsets of the line. An affine copy of a set $A\subset\mathbb R$ has the form

$$x+sA=\{x+sa:a\in A\},\qquad s\ne0.$$

The number $x$ translates the pattern. A positive $s$ changes its size, while a negative $s$ also reflects it. Excluding $s=0$ prevents collapsing the whole pattern to one point.

A set is measure universal if every measurable set of positive measure contains such a copy. Erdős’s conjecture says that no infinite set is measure universal.

For a fixed $q\in(0,1)$, write

$$G_q=\{q^n:n=1,2,3,\ldots\}.$$

This sequence approaches zero, but zero itself is not included. The distinction becomes useful in the construction: a neighborhood of a copy’s limiting point will still contain sufficiently late members of the copy.

Why are finite patterns universal? If $F$ is finite and $A$ has finite positive measure, sufficiently small translations change $A$ only a little in measure. Consequently the intersection of the finitely many sets $A-sa$, for $a\in F$, still has positive measure when $s$ is small enough. A point $x$ in that intersection satisfies $x+sF\subset A$. There is no comparable automatic conclusion for infinitely many intersections.

3. Development history and previous partial results

Erdős formulated the similarity problem in 1974. Falconer’s On a problem of Erdős on sequences and measurable sets (1984), followed by Eigen’s work in 1985, treated positive decreasing sequences whose successive ratio tends to one. Those sequences approach zero much more slowly, in relative terms, than a fixed geometric sequence.

Humke and Laczkovich developed further criteria through finite covering problems. Kolountzakis’s Infinite patterns that can be avoided by measure (1997) introduced a probabilistic criterion involving large finite subsets with sufficiently well-spaced points. The account by Chlebík presents a translation-invariant form: roughly, the smallest gap, after normalizing the diameter, must not shrink exponentially fast in the number of selected points. Geometric progressions fail that test precisely because their gaps shrink exponentially.

Additive structure gave another route. Bourgain proved in 1987 that the sum of three infinite sets is nonuniversal. Later results treated certain two-fold sums, including a sum of two dyadic sequences. A 2026 manuscript on geometric summands extended this direction. Avoiding a sumset, however, does not imply avoiding each smaller pattern contained inside it: the latter is the harder requirement.

Other notions of size and allowed transformations lead to different answers. Cruz, Lai and Pramanik constructed large-dimensional avoiding sets, with the relevant dimension-one subsets having zero Lebesgue measure. Feng, Lai and Xiong’s bi-Lipschitz embedding result allows geometric sequences to embed in every positive-measure set when the map may distort individual distances within fixed factors. Affine maps are much more rigid.

Against this background, M2, dated September 25, claims the dyadic affine case. M1, dated October 5, extends the construction to every fixed $q\in(0,1)$. The order of those dates explains the relationship between the two papers: the dyadic construction is a special case of the later, broader statement.

4. What the new manuscripts claim

M1 states that for every fixed $q\in(0,1)$ and every $\eta\in(0,1)$, there is a compact set $E_{q,\eta}\subset[0,1]$ such that

$$m(E_{q,\eta})>1-\eta$$

and, for every $x\in\mathbb R$ and $s\ne0$, at least one positive integer $n$ satisfies

$$x+sq^n\notin E_{q,\eta}.$$

Thus the set misses at least one point from every affine copy of the entire infinite sequence. Both signs of the dilation are included. No restriction is placed on the location of the limiting point $x$.

M2 gives the same conclusion with $q=1/2$. The general paper retains nested dyadic grids in its construction, but adapts their resolutions to the chosen geometric ratio. Its parameters, and the resulting avoiding set, may depend on $q$.

The statement concerns avoidance of a complete infinite copy. It does not say that the set avoids long finite geometric patterns. In fact, because its measure is positive, it must contain affine copies of every fixed finite truncation. The translation and scale needed for those finite copies may change as the truncation grows.

5. Why the advance matters

Importance: 5/5. The geometric case is a central and longstanding obstruction in the Erdős similarity problem. A method that handles exponentially shrinking gaps would go beyond the earlier spacing criteria and resolve one of the most prominent test cases of the conjecture.

The near-full-measure conclusion makes the phenomenon especially striking. For any prescribed small percentage, the construction claims that one can remove less than that percentage of the unit interval and destroy every affine copy of a chosen infinite geometric sequence. Positive measure guarantees every finite pattern, yet even measure arbitrarily close to one need not guarantee this particular infinite pattern.

The distinction is about exact configurations, not numerical approximation. A set may contain points extremely close to an entire progression without containing the progression itself. Small holes arranged at many scales can exploit that difference.

The probabilistic mechanism also has independent conceptual interest: it controls an uncountable range of translations and dilations using finite random choices and a carefully organized hierarchy. The score reflects this significance if the manuscript claims hold; the full conjecture for arbitrary infinite sets remains outside their stated conclusion.

6. The main idea

Build the holes first

Instead of directly describing a large avoiding set, construct a small open set that intersects every forbidden copy. Its complement inside $[0,1]$ is then compact and has large measure.

The first target is normalized: build a one-periodic open set $H$ of very small density that meets every $x+tG_q$ with $t\in[1,2]$. Periodicity allows translation to be studied on one unit interval. Scaled and reflected copies of such sets, with summable measure budgets, then cover all positive and negative dilation sizes.

Route each point through a finite tree

A naive random selection of tiny intervals gives many opportunities to hit a fixed copy. Its weakness is the number of possible scales: inspecting every possible change on the finest grid may cost more than the random estimates can afford.

The manuscripts instead use a finite ordered tree. Independent binary tables on grid cells route a point to a leaf. At each branch the first successful child is chosen; if all tested children fail, the final child is used by default. A separate random table at the leaf selects membership in the hole set with a small probability $p$. This keeps the expected total density small.

Different edges receive separated windows of sequence indices. The gaps between windows make earlier routing decisions stable under the later, smaller translations, except near a small set of grid boundaries. At the first default on a point’s route, the unused children provide many fresh opportunities for translated sequence points to hit the selected set.

Make the scale count local

The decisive organization concerns which grids a particular group of tests needs. Its index window is chosen long enough relative to its following gap and descendant subtree. The number of possible dilation patterns for that group can then be counted using its own portion of the construction, rather than the finest grid anywhere in the tree.

Independent tests give a rapidly decreasing probability of missing every opportunity. The local scale count is arranged to grow slowly enough for that probability to dominate it. For general $q$, dyadic grid resolutions are chosen comparable to the geometric separation at the relevant indices; the branching number is allowed to depend on the ratio.

Finally, some centers may remain exceptional. Their set has small measure and can be covered by a small open neighborhood. If a copy converges to a center inside this neighborhood, sufficiently late points also lie in it. Adding these neighborhoods repairs the remaining copies without exhausting the measure budget.

7. Scope, limitations, and open questions

The ratio $q$ is fixed before the avoiding set is constructed. The paper expressly makes no simultaneous assertion for all ratios. Constants and choices can depend on $q$, especially when its geometric spacing changes substantially.

The full Erdős similarity conjecture concerns every infinite subset of the line, not just a geometric null sequence. That larger problem is not resolved by the theorem stated here.

Affine avoidance also differs from avoidance under more flexible maps. A bi-Lipschitz image may fit even where no affine image does. Likewise, Hausdorff dimension one is weaker than positive Lebesgue measure for this question; earlier dimension results do not supply the present measure conclusion.

The construction is an existence argument using finite random choices and an infinite assembly of scale budgets. It does not provide a simple interval-removal recipe with practical numerical parameters. Any resulting set has no nonempty interval in its interior, since an interval contains small affine copies of the sequence. Its large measure therefore coexists with persistent small-scale gaps.

8. Sources and further reading

M1 supplies the all-ratios theorem, the historical distinctions and the local scale-count mechanism. M2 presents the earlier dyadic construction. Their introductions and normalized hitting-set propositions are the main sources for this explanation.

Chlebík’s linked paper provides a useful primary account of the older spacing criteria. The linked papers on sumsets, large-dimensional avoidance and bi-Lipschitz embeddings explain why several superficially similar partial results leave the fixed geometric affine problem untouched.

Source manuscripts

• M1. OpenAI, The geometric case of the Erdős similarity conjecture (October 5, 2026).
• M2. OpenAI, The dyadic case of the Erdős similarity conjecture (September 25, 2026).

Sources consulted on 14 October 2026 at repository revision 1581d4c. Download the manuscript citations.