Large subsets avoiding algebraic patterns

We prove the existence of a subset of the torus with large sumsets and avoiding all linear patterns. This extends a result of Körner, who had shown that for any integer $q \geq 1$, there exists a subset $K$ of $\mathbb R/\mathbb Z$ satisfying no non-trivial linear relations of order $2q-1$ and such that $q.K$ has positive Lebesgue measure. Our method is based on transfinite induction, which also allows us to produce large sets in different senses (cardinality, outer Lebesgue measure or Hausdorff dimension) avoiding families of algebraic patterns, for example Sidon sets in infinite abelian groups with small $2$ and $3$-torsion or sets with no repeated distances in $\mathbb R^n$. We also discuss questions of measurability of such sets and the role of the axiom of choice in our constructions.

💡 Research Summary

The paper investigates the simultaneous achievement of two seemingly antagonistic goals: constructing large subsets of the torus (or more generally of abelian groups) that avoid a wide class of algebraic patterns, while ensuring that their sumsets are “large” in the sense of Lebesgue measure, outer measure, or Hausdorff dimension. The authors extend Körner’s classical result, which produced a closed set E ⊂ ℝ/ℤ avoiding non‑trivial solutions to a single linear equation of order 2q‑1 and guaranteeing that q·E has positive measure.

The central notion introduced is that of an independent set: a subset A of an abelian group G that contains no non‑trivial solutions to any integer‑coefficient linear equation. Trivial solutions are precisely those forced by the algebraic structure (e.g., all variables equal when the sum of coefficients is zero). Lemma 1.4 shows that, given a proper set A, the collection of elements that would create a non‑trivial solution is bounded in cardinality by a function of |A| and the sizes of kernels of multiplication maps. This quantitative bound is the engine of the transfinite construction.

Using the axiom of choice (in a strong form), the authors well‑order the family of perfect closed subsets of the torus, indexed by the continuum cardinal 𝔠. By transfinite induction they build an increasing chain (Kα)α<𝔠 such that each Kα remains independent, stays disjoint from the corresponding perfect set Pα, and has size at most |α|+1. At limit stages the union preserves independence because the set of forbidden elements remains of size <𝔠. The final set K=⋃α<𝔠 Kα has outer Lebesgue measure 1; its complement is a countable union of isolated points, hence of measure zero. Consequently every dilate q·K also has positive outer measure for any integer q≥1. The construction necessarily yields a non‑measurable set (or a non‑measurable sumset), as shown in Lemma 6.3, indicating that a stronger choice principle than Dependent Choice is required.

The paper then turns to Sidon sets, i.e., subsets S of an abelian semigroup for which the equation a+b=c+d implies {a,b}={c,d}. Theorem 3.1 treats an infinite abelian group G of cardinality κ with the property that the kernels of the maps x↦2x and x↦3x are strictly smaller than κ. Assuming a set E⊂G such that each element of G has exactly κ representations as a sum of two elements of E, the authors again employ transfinite induction. At each stage they add a carefully chosen pair (aβ,bβ)∈E that avoids a finite “obstruction” set Oβ constructed from previously chosen elements. The obstruction set is shown to have size <κ thanks to the kernel hypotheses. The resulting set S⊂E is Sidon and satisfies S+S=G, i.e., its sumset covers the whole group. This dramatically strengthens earlier results on maximal Sidon sets by showing that a Sidon set can have a full sumset while still being “large” in cardinality.

In Section 4 the authors explore the relationship between Hausdorff dimension and the measure of sumsets. They prove that there exist sets E⊂