Minimal order-two bases and quantitative thinning

Definitions and mathematical statement

Use ${\mathbb{N}}_0=\{0,1,\ldots\}$ for a precise convention. A set $A\subseteq{\mathbb{N}}_0$ is an exact order-two basis if $A+A={\mathbb{N}}_0$, and an asymptotic order-two basis if $[N_A,\infty)\cap{\mathbb{Z}}\subseteq A+A$ for some $N_A$. Repeated summands are allowed. A basis is minimal in either sense if no proper subset remains a basis in that same sense. The source asks whether every order-two basis contains a proper basis subset, and how slowly a resulting counting function $B(x)=|B\cap[0,x]|$ can grow. It does not distinguish exact from asymptotic basis or whether zero belongs to the naturals.

The attached aid reports that the universal proper-subset assertion fails under either interpretation: minimal asymptotic bases are classical, including thin strongly minimal examples, while every exact basis has a minimal exact subbasis by finite-representation compactness. Starting with such a minimal basis leaves no proper admissible subset. Thus a question about one prescribed $A$ differs from optimizing growth over all bases.

For every fixed order-two basis, pair counting yields \[ B(x)\ge\sqrt{2x}-O(1)\quad(x\to\infty). \] The aid reports constructions with $B(x)=\Theta(\sqrt x)$, so the square-root order is globally optimal. This does not claim an optimal leading constant or a thin proper subbasis of every prescribed $A$.

Its quantitative sufficient thinning condition uses unordered representations \[ r_A(n)=\#\{(a,b):a,b\in A,\ a\le b,\ a+b=n\}. \] For $F\subseteq A$, if \[ r_A(n)>|F\cap[0,n]|\qquad\text{for all sufficiently large }n, \] then $A\setminus F$ remains an asymptotic basis, since each removed element can spoil at most one unordered representation of a fixed $n$. In particular if $r_A(n)\to\infty$, choose increasing thresholds $N_j$ beyond which $r_A(n)>j$, and increasing $f_j\in A$ with $f_j\ge N_j$. The aid states that $F=\{f_j:j\ge1\}$ is an infinite removable set. This hypothesis is convergence to infinity, not merely unboundedness, and ensures an asymptotic conclusion rather than preservation of every initial exact representation.

Short English statement

Minimal bases refute universal proper thinning. The supplied aid separates that obstruction from the globally optimal square-root counting scale and gives a representation-surplus condition guaranteeing an infinite removable subset for some asymptotic bases.

Sources

• Local UnsolvedMath registry, permanent ID $20001110$, source code catalogue entry; canonical aim-combinatorics-notes.json, notes[234].
• American Institute of Mathematics, Recent trends in additive combinatorics, problem 2.10: https://aimath.org/WWN/additivecomb/additivecomb.pdf. The supplied verification reports agreement between the official PDF and corpus wording; no PDF inspection is claimed here. Original literature and remarks fields are empty.
• Attached research attempt 1: counterexample, recorded run status refuted, rigor proved, novelty confidence low; registry status solved, classification SOLVED-IN-LITERATURE. It calls the minimal-basis and thin-construction facts classical but gives no complete references. Its machine-generated, non-peer-reviewed claims require independent verification before citation.