Finite real sets have dissociated subsets of size log_2 n - O(log log log n)

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_proof: The bounded-ratio bound L212 in each dyadic block of the absolute values, a doubling chain from alternate blocks above the chosen one, and a weighted choice of that block.

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Statement

For a finite set $X\subset\mathbb R$ let $d(X)$ be the largest size of a dissociated subset. For every finite $A\subset\mathbb R$ with $n=|A|\ge5$,

$$d(A)\ \ge\ \log_2(n-1)-\log_2\!\bigl(\lceil\log_2\log_2n\rceil+2+\sqrt2\bigr)-\log_2 6 .$$

In the notation of Problem 963 this gives $f(n)\ge\log_2n-\log_2\log_2\log_2n-4$ for every $n\ge2^{16}$, that is, $\log_2n-f(n)\le\log_2\log_2\log_2n+4$.

Significance

The tier-2 bound [[theory/number_theory/L22_near_logarithmic_dissociated_subset/_index|L22 loses $\log_2(\log_2n+1)$ against $\log_2n$]]. This claim lowers the order of the loss to $\log_2\log_2\log_2n$, but its constants are worse: it is the stronger bound only for sufficiently large $n$. It does not decide whether $f(n)\ge\lfloor\log_2n\rfloor$, and it does not give $f(n)\ge\log_2n-O(1)$.

Argument

The proof, by Plasma AI, is author-recorded at tier 0. It has no independent review and no formalization.

Dependency roles. The proof applies L212, which gives each dyadic block of n_i positive reals a dissociated subset of size greater than log_2 n_i - log_2 3, to every dyadic block of the absolute values. It extends the subset found in one block by a doubling chain drawn from alternate occupied blocks far enough above it, and chooses the block by a weighted pigeonhole. The loss of order $\log_2\log_2\log_2n$ comes from the blocks immediately above the chosen one: they lie within a factor of order $\log_2n$ of its elements, so a doubling chain cannot use them.

The editable proof sketch of the sharp-bound attack records this combination step as the remaining gap toward a constant loss for arbitrary sets.

Verification and remaining obligations

Read the proof page together with L212's. The steps are the symmetric reduction, the dyadic blocks, the doubling-chain lemma, the weighted pigeonhole and the explicit simplification for $n\ge2^{16}$. No mathematical gap is known in this statement beyond the standing of L212, which is also author-recorded. Independent review has not been commissioned.