Short··0:20·Real and complex analysis
Erdős Similarity Conjecture: OpenAI Claims the Geometric Case
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About this video
Erdős asked whether every infinite set of reals can be dodged by some set of positive measure: a set containing no shrunk, shifted or flipped copy of the pattern. For geometric sequences 1, q, q², … OpenAI claims a proof: for every ratio q between 0 and 1 there is a compact set filling as much of [0, 1] as you like that contains no affine copy of {qⁿ}, for either sign of dilation (the set may depend on q). These are OpenAI's preprints, not yet peer reviewed.
Papers (OpenAI, catalogue result 084):
- The geometric case of the Erdős similarity conjecture: github.com/openai/math/blob/main/preprints/The-geometric-case…
- The dyadic case of the Erdős similarity conjecture: github.com/openai/math/blob/main/preprints/The-dyadic-case-of…
The animation illustrates the result; it is not a proof.
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Every result explained: gradientdamage.com/p/084
About the problem
Real and complex analysis·OpenAI Math·Result 084
The geometric case of the Erdős similarity conjecture
For every fixed q ∈ (0,1), constructs compact subsets of [0,1] with measure arbitrarily close to one containing no translated and nontrivially dilated copy of {qn:n ≥ 1}, with dilations of either sign. This resolves the geometric-progression case of the Erdős similarity conjecture for every ratio. (Lean)
Topics
Erdős similarity conjectureErdős similarity problemErdos similarityPaul Erdősgeometric sequencegeometric progressiondyadic sequenceLebesgue measuremeasure theoryreal analysisaffine copyKolountzakisSteinhaus problemOpenAIOpenAI mathAI mathematicsAI proofmathsmathematicsmath shorts3D animationErdos similarity conjectureErdos similarity problemPaul Erdos
