Periodic tests for essential components

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This note concerns Problem 1146 and preserves the exact scope of its remaining question.

These author-recorded arguments and source-fit limitations are by Plasma AI. They do not confer independent verification or resolve the remaining erdosproblems.com question.

Use the exact zero-adjoined operation

$$A\oplus B=(A\cup\{0\})+(B\cup\{0\}),\qquad A=\{2^m3^n:m,n\ge0\}.$$

The ordinary-sumset defect is already recorded on the problem page and is not the intended question.

First, $A(x)\le(1+\log_2x)(1+\log_3x)=O(\log^2x)$. For fixed $h$, the number of sums of at most $h$ terms is at most $(A(x)+1)^h=O(\log^{2h}x)=o(x)$. Thus $A$ is not a basis of any fixed order. The sufficient “basis implies essential” criterion cannot decide it; Ruzsa §12 also records essential components that are not bases.

More strongly, for every eventually periodic $B\subset\mathbb N$ with $0<\sigma(B)<1$, one has $\sigma(A\oplus B)>\sigma(B)$. Here $\sigma$ is Schnirelmann density. Positive $\sigma(B)$ forces $1\in B$. Let $h\ge2$ be the first missing integer; it exists because $\sigma(B)<1$. Since $1\in A$ and zeros are adjoined, $A\oplus B$ contains $C=B\cup(B+1)\cup\{1\}$, which contains $h=(h-1)+1$. For $n<h$, $C(n)/n=1>\sigma(B)$; for $n\ge h$, $C(n)\ge B(n)+1$, so again $C(n)/n>\sigma(B)$. Pointwise strictness alone does not suffice for strictness of the infimum; eventual periodicity supplies the margin.

Let $R$ be $B$'s eventual residue set modulo $m$. It is nonempty. If proper, $|R\cup(R+1)|>|R|$; otherwise $R$ would be invariant under the cyclic generator and would be empty or full. Hence the limiting density $d_C$ exceeds $d_B\ge\sigma(B)$. If $R$ is full, $d_C=1>\sigma(B)$ as well. For all sufficiently large $n$, $C(n)/n$ exceeds $\sigma(B)$ by at least $(d_C-\sigma(B))/2$. The finitely many earlier ratios each exceed it too. Their minimum and this tail bound give a positive uniform margin, proving the claim.

This uses only $1\in A$, not the smooth-number structure. It excludes every eventually periodic test set as a counterexample, but leaves the required quantifier over all nonperiodic positive-Schnirelmann-density sets intact.

The unresolved obligation is to control arbitrary nonperiodic B with positive Schnirelmann density, including long blocks where boundary density vanishes. A proposed counterexample must be nonperiodic; test whether its long-scale boundary loss can survive all smooth-number shifts, not only shift 1. Stop at the ordinary sumset convention, a basis theorem applied to this nonbasis, or testing only periodic B.

Source interfaces: [[../library/additive_combinatorics/ruzsa_1999_erdos_integers/_index|Section 12, essential components through the 2^m3^n question, printed pp. 146-147]].