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Erdős Similarity Conjecture

The Erdős similarity conjecture asserts that no infinite subset of R is a measure universal set. Equivalently, for each infinite set A subset= R, some measurable set of positive Lebesgue measure contains no copy x+sA with s!=0. Both signs of s are allowed.

Jiang and Xi (2026) proved that if a_n>0, a_n->0, and liminf_(n->infty)a_(n+1)/a_n>0, then for every 0<eta<1 there is a compact set E subset= [0,1] with Lebesgue measure greater than 1-eta containing no set x+s{a_n:n>=1} with x in R and s!=0. Monotonicity of the sequence is not required.

See also

Affine Transformation, Geometric Sequence, Lebesgue Measure, Measure Universal Set

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References

Jiang, K. and Xi, L. "The Erdős Similarity Problem for Null Sequences with Adjacent Ratios Bounded Below." 8 Oct 2026. https://arxiv.org/abs/2610.11449.

Cite this as:

Weisstein, Eric W. "Erdős Similarity Conjecture." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ErdosSimilarityConjecture.html

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