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The deductions and obstructions here are author-recorded by Plasma AI. Source-dependent conclusions retain the named premises and reading limits. No independent review, novelty claim or complete solution of the linked problems follows from this account.

The relevant questions are E0953, E0959, E1084.

Quantitative construction interfaces

E0953: the finite clearance objective must retain area

Write $N(r,\delta)$ for the largest size of a finite set in the radius-$r$ disk whose distinct-center lengths are all at distance at least $\delta$ from every integer, including zero. Let $0<\delta<1/10$. Around a set $P\subset B_{r-1}$ with this clearance, put open disks of radius $\delta/4$. They are disjoint because distinct centers are at least $\delta$ apart. They lie inside $B_r$. Within each small disk all distances are below one. Between two disks the change from the center distance is less than $\delta/2$, so every positive integer is still avoided. The area is exactly

$$\frac{\pi}{16}\delta^2|P|.$$

Conversely, take a compact subset $K$ of any admissible measurable set in $B_r$. Its distance image is compact and avoids the finitely many positive integers in its range, hence has a positive clearance $\eta$. For $0<\delta<\min(\eta,1/10)$, a maximal $\delta$-separated net $P\subset K$ is finite, covers $K$ by radius-$\delta$ disks, and satisfies the clearance condition also at zero. Thus $|K|\le\pi\delta^2|P|\le\pi\sup_\delta\delta^2N(r,\delta)$. Inner regularity of Lebesgue measure gives the same upper bound for the area supremum. This is the robust reduction in the inspected proof of Chojecki's Lemma 2, with an explicit lower-bound constant and radius shift.

To prove the requested $M(r)\gg\sqrt r$, a finite construction must achieve $\delta^2|P|\gg\sqrt r$. A finite set with noninteger distances always has some positive clearance, but that clearance can be too small for the desired area. The next test must quantify it. The source's kernel construction and stronger quantitative claims were not re-audited; the exponent-level result is not mistaken for the missing constant-factor result.

E0959: a fully controlled $n\log n$ gap

Choose $t\ge2$ unit vectors $v_1,\ldots,v_t$ in generic planar directions and form all subset sums $p_\varepsilon=\sum_i\varepsilon_i v_i$, $\varepsilon\in\{0,1\}^t$. There are $n=2^t$ distinct points for generic directions. A nonzero difference has coefficient vector $c\in\{-1,0,1\}^t$, with squared norm

$$\sum_i c_i^2+2\sum_{i<j}c_ic_j\cos(\theta_i-\theta_j).$$

The distinct nonconstant Fourier characters and the constant term are linearly independent. Consequently two such norm polynomials are identical only if both supports are singletons, or if the support has at least two entries and the vectors agree up to an overall sign. Indeed the off-diagonal products $c_ic_j$, together with support size, determine the support and all relative signs. All other equalities, and all collisions, are nonzero analytic equations in the angles. Their finite union has empty interior, so they can all be avoided simultaneously.

A signed difference with support size $s$ then occurs in $2^{t-s}$ unordered pairs: the coordinates outside the support may be chosen freely, and reversing the pair supplies the negative difference rather than an additional unordered pair. All $t$ singleton supports merge into the unit class, giving multiplicity $t2^{t-1}=tn/2$. Every other class has multiplicity at most $2^{t-2}=n/4$, and support two attains that bound. Hence

$$M(n)\ge \frac n2\log_2 n-\frac n4\qquad(n=2^t,\ t\ge2).$$

Generic padding to any larger order below $2n$ adds only singleton new lengths: choose each new point off the finitely many old-distance circles and perpendicular bisectors. This proves an $\Omega(n\log n)$ gap for all sufficiently large orders.

This direct specialization is consistent with the inspected proof of Clemen--Dumitrescu--Liu Theorem 1.9 and Corollary 1.10 in Section 4. It is not a new matching upper bound or an acceptance of stronger pending claims. A construction with many unit pairs does not by itself bound its second-richest distance. Any attempted improvement must inspect that competitor explicitly.

E1084: extend the special-size packing coefficient to all sizes

Normalize minimum center distance to one, and let $f(n)=f_3(n)$ count unordered contacts. The inspected statement recalled on page 2 of Bezdek--Reid gives, with

$$c=486^{1/3}=6(3/2)^{2/3},\qquad n_k=(2k^3+k)/3,$$

the strict inequality $f(n_k)>6n_k-cn_k^{2/3}$ for $k\ge2$. Their unit-radius-ball convention has contact distance two; scaling the centers by one half gives this normalization with the contact count unchanged. The earlier construction proof cited there was not inspected. The following interpolation is conditional only on that displayed source statement.

Distant union gives $f(a+b)\ge f(a)+f(b)$, and adding distant isolated points gives monotonicity. For $n\ge6$, take the largest $k$ with $n_k\le n$, and set $r=n-n_k$. Then

$$r<n_{k+1}-n_k=2k^2+2k+1\le4k^2 \le4(3n/2)^{2/3}.$$

Monotonicity first yields

$$f(n)>6n-6r-cn^{2/3}\ge6n-5cn^{2/3}.$$

The weak inequality $f(n)\ge6n-5cn^{2/3}$ holds also for $0\le n\le5$, because the right side is nonpositive; put $f(0)=0$. Fill the residual with this uniform construction, instead of isolated points. Superadditivity now gives

$$\begin{split} f(n)&>6n-cn_k^{2/3}-5cr^{2/3}\\ &\ge6n-cn^{2/3}-C n^{4/9},\\ C&=5c\,4^{2/3}(3/2)^{4/9},\qquad n\ge6. \end{split}$$

Thus the coefficient $c$ in the construction's $n^{2/3}$ deficit extends asymptotically to every size. This is a completed conditional deduction with an explicit error term, not a claim of novelty. The inspected upper theorem is $f(n)<6n-0.926n^{2/3}$, so the boundary constant remains between 0.926 and $c=7.862\ldots$. Neither the exact function in dimension three nor the higher-dimensional question is settled. The theorem and recalled lower bound were checked against PDF page 2; no transcription correction was needed.

Source interfaces

These source locators identify the uses in this account. Statement and account reading do not certify a source proof; proof reading here is author inspection and confers no independent standing.

• [[../library/distance_problems/bezdek_2013_contact_graphs_unit_sphere_packings_revisited/_index|bezdek 2013 contact graphs unit sphere packings revisited]]: Section 1, Theorem 1 and immediately following recalled lower bound, PDF p.2 (statement reading).
• [[../library/distance_problems/chojecki_2026_order_growth_planar_sets_avoiding_integer/_index|chojecki 2026 order growth planar sets avoiding integer]]: Section 1, Lemma 2 and its proof (proof reading). Section 1, Theorems 1 and 3 (statement reading).
• [[../library/distance_problems/clemen_2025_multiplicities_interpoint_distances/_index|clemen 2025 multiplicities interpoint distances]]: Section 4, proof of Theorem 1.9 and deduction of Corollary 1.10 (proof reading). Theorem 1.9; Corollary 1.10; Problem 1.11 (statement reading).