A superlacunary test for similarity criteria

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This author-recorded argument by Plasma AI concerns Problem 120 at the formulation and standing recorded there. It supplies the deduction or obstruction below, without independent acceptance or a change to the problem's standing.

Let $A=\{2^{-2^j}:j\ge1\}$. The survey's Eigen–Falconer theorem needs a decreasing sequence whose ratios tend to one; the problem account's newer ratio theorem still needs ratios bounded away from zero. For every subsequence indexed by $j_s$,

$$\frac{2^{-2^{j_{s+1}}}}{2^{-2^{j_s}}} \le 2^{-2^{j_s}}\longrightarrow0.$$

An affine image has the same ratios of distances from its unique accumulation point. Thus neither passing to a subset nor applying a nonzero affine scale puts this example into either ratio class. Scales of both signs are included.

The Bourgain three-sum theorem quoted in the survey also cannot enter through a sumset subset. Every positive difference in $A$ is unique: for $i<j$,

$$2^{-2^i}-2^{-2^j} =\frac{2^{2^j-2^i}-1}{2^{2^j}}.$$

Its numerator is odd, so its reduced denominator determines $j$ and then its numerator determines $i$. If $B+C\subset A$ with each of $B,C$ containing two different elements, the two different ordered pairs $(b_1+c_1,b_2+c_1)$ and $(b_1+c_2,b_2+c_2)$ have the same positive difference (order $b_1<b_2$). This contradicts uniqueness, even when the middle entries coincide. A sum of three infinite sets contains such a translate of a two-by-two sum, so that route does not cover $A$.

A Fourier-decaying probability-measure route cannot cover this countable set either. For a probability $\mu=\sum p_j\delta_{a_j}$ supported on a countable set, averaging $|\widehat\mu(\xi)|^2$ over $[-T,T]$ and using dominated convergence on the absolutely summable double series gives the limit $\sum p_j^2>0$. If $\widehat\mu(\xi)\to0$ as $|\xi|\to\infty$, that average would tend to zero. The same argument applies to the countable closure of $A$.

These are complete obstructions to the specified hypotheses, not an affine-universality counterexample or a disproof of the similarity conjecture. The Eigen–Falconer and Bourgain inputs were read in held result accounts of the survey, not in the original proofs. The recent manuscript criteria are known here through the problem account, rather than fresh inspection of their proofs. A discriminating continuation: find a non-universality criterion which actually admits this superlacunary, unique-difference, atomic example. Stop on any criterion still requiring bounded ratios, an infinite sumset, or a non-atomic Fourier-decaying measure.

Source interfaces

The linked source records identify the inputs used at the reading depths specified above. A result account is not a check of the source's proof.

• theorem 1 3.
• theorem 1 4.