A Regularly Thin Minimal Asymptotic Basis of Order Two

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theorem_1_1: Bhalla's manuscript theorem that for some constant C > 0 there is a minimal asymptotic basis A of the natural numbers of order 2 whose counting function is C sqrt(x) + O(1), so that its kth element is asymptotic to C^(-2) k^2.

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Aron Bhalla, A Regularly Thin Minimal Asymptotic Basis of Order Two. unpublished manuscript (2026).

Theorem 1.1 asserts the existence of a constant C > 0 and a minimal asymptotic basis A of the natural numbers of order 2 with A(x) = C sqrt(x) + O(1), consequently a_k ~ C^(-2) k^2 for the increasing enumeration of A (p. 1). This is the Cassels-type regularity question for minimal bases: earlier thin minimal bases (powers-of-two and digital partition constructions, cited to Nathanson 1988, after the early work of Stohr and Hartter on minimal bases) achieve the right order x^(1/2) but with block-structured, oscillatory counting functions, not the exact square-root regularity here. The construction keeps a single private witness active at a time. In epoch j it sets aside a large integer X_j to become the private witness of an element a_j placed earlier; ordinary quotient-remainder blocks then cover the following stretch of sums, none of them representing X_j; a lower seam is inserted just below X_j, then the release element b_j = X_j - a_j is added so that X_j = a_j + b_j is its unique representation, smoothing fills the square shells to prescribed capacities, the prefix up to X_j is permanently closed, and an upper seam handles the boundary near 2X_j (pp. 2, 12-16). The finite ingredients are a separating code (Lemma 2.1, pp. 2-3), the external-protected block lemma (Lemma 3.2, p. 5), a symmetric boundary seam (Lemma 4.1, p. 7), shell and smoothing estimates (Lemmas 5.1-5.3, p. 9) and forced-shell bounds (Lemma 6.3, p. 11), with Section 8 (pp. 16-17) verifying the basis, counting and minimality properties. For problem 326 this is the primary manuscript: if Theorem 1.1 holds, a_k/k^2 tends to the nonzero constant C^(-2), which answers the question affirmatively. The manuscript is unpublished and unrefereed, and it does not mention a formalization.

Source: https://drive.google.com/file/d/1VKaFmiMWWMW7NME-L47HVGSWOoBt9sug/view. No notice is printed on the first or last two pages; the copy read for this card is the author's unpublished manuscript, shared by a Google Drive link (https://drive.google.com/file/d/1VKaFmiMWWMW7NME-L47HVGSWOoBt9sug/view) that states no terms, and no publisher page exists; the term is unstated.

Bears on. #326: Theorem 1.1 asserts a minimal basis of order 2 with $a_k/k^2\to C^{-2}\neq0$; if the theorem holds, it answers the problem's question yes. The manuscript is unrefereed.

Results.

• Theorem 1.1 (p. 1): for some constant $C>0$ there is a minimal asymptotic basis $A\subset\mathbb N$ of order $2$ with $A(x)=C\sqrt x+O(1)$, so $a_k\sim C^{-2}k^2$; the page also records the paper's minimality criterion (p. 1), that $A$ is minimal when removing any one element destroys infinitely many sums.

The finite lemmas are tools of the construction only and get no pages of their own: Lemma 2.1 (pp. 2-3), for every $M\ge1$, gives subsets $\Gamma_1,\ldots,\Gamma_M$ of $\{1,\ldots,R_0(M)\}$ such that for all disjoint $I,J\subset\{1,\ldots,M\}$ with $I\cup J\neq\varnothing$ some $\lambda$ lies in $\Gamma_j$ for every $j\in J$ and in no $\Gamma_i$ with $i\in I$; Lemma 3.2 (p. 5), with $B$, $D$ depending only on $M$ and for all sufficiently large $L$, covers an interval $J=[N,N+L)$ outside a protected set $P\subset J$ by $S+S$ while keeping $P\cup Q$ out of $S+S$, for $\lvert P\rvert+\lvert Q\rvert\le M$ and $Q\cap J\subset P$, avoiding a forbidden set of at most $C_F\sqrt L$ points, with $\lvert S\rvert\le B\sqrt L$ and $S\subset[N/2-DL,N/2+DL]$; Lemma 5.3 (p. 9), when every square shell holds at most $C_*$ points, fills a sufficiently large shell below $X$ up to a count $q_r=O_C(1)$ without a second representation of a uniquely represented $X$; and Lemma 6.5 (p. 12) keeps a unique representation of $n$ unique when no element at most $n$ is added later.

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