Density versions of Plünnecke inequality

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lemma_1: Converts a minimal forward density on an integer interval into the Plünnecke lower bound for the sum with a Schnirelmann basis.

theorem_2: Proves the Schnirelmann density bound for a sum with a basis using Jin's interval partition and the external truncated Plünnecke inequality.

theorem_3: States the external monotonicity theorem for magnification ratios in a truncated additive graph, with the proof references supplied by Jin.

theorem_4: States Jin's lower asymptotic density inequality and sketches the finite trimming argument that replaces Schnirelmann density of the iterated sum.

theorem_5: Records Jin's externally proved counterexample to the direct upper asymptotic density analog of the Plünnecke basis bound.

theorem_6: States the upper Banach density form of Jin's inequality and sketches the long-interval argument for its finite graph input.

theorem_7: States Jin's mixed lower and upper Banach density inequality and sketches its uniform-interval proof.

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Renling Jin, “Density Versions of Plünnecke Inequality: Epsilon-Delta Approach,” in Combinatorial and Additive Number Theory, Springer, 2014, 99–113, DOI 10.1007/978-1-4939-1601-6_8. The publisher-deposited Crossref record dates online publication to 26 September 2014.

Version used. The copy read for this card is the sixteen-page author manuscript downloaded from Jin's website on 5 September 2026. Its PDF metadata gives 19 September 2011; that is a file metadata date, not a verified revision or publication date. That copy has 194385 bytes. Its printed pages 1–16 agree with PDF pages 1–16. All result labels and proof locators below refer to this manuscript. The published chapter was identified bibliographically but its text was not compared, so equivalence of versions and labels is not asserted. The manuscript read is the author's copy from the author's site (https://jinr.people.charleston.edu/), which states no terms, and it prints no notice; the term is unstated.

For $C\subseteq\mathbb N_0=\{0,1,\ldots\}$, write $C(a,b)=|C\cap[a,b]|$, where intervals have integer endpoints, and $C(n)=C(1,n)$. The densities used are

$$\sigma(C)=\inf_{n\geq1}\frac{C(n)}n,\qquad \underline d(C)=\liminf_{n\to\infty}\frac{C(n)}n,\qquad \overline d(C)=\limsup_{n\to\infty}\frac{C(n)}n,$$ $$\underline u(C)=\lim_{n\to\infty}\inf_{a\in\mathbb N_0} \frac{C(a,a+n)}{n+1},\qquad \overline u(C)=\lim_{n\to\infty}\sup_{a\in\mathbb N_0} \frac{C(a,a+n)}{n+1}.$$

The last two are lower and upper Banach density on the nonnegative integers. For a positive integer $h$, $hB$ is the sum of exactly $h$ elements of $B$. Jin calls $B$ a Schnirelmann basis of order $h$ when $hB=\mathbb N_0$. This forces $0\in B$; it therefore also permits representations with at most $h$ summands by padding with zeros. Zero is excluded from the counting function defining $\sigma$, but is not automatically adjoined to either summand of a sumset.

Main proof. [[additive_bases/jin_2014_density_versions_plunnecke_inequality/theorem_2|Theorem 2]] states Plünnecke's bound $\sigma(A+B)\geq\sigma(A)^{1-1/h}$ and gives Jin's complete rewritten argument from Section 4, pp. 14–15. The same-paper [[additive_bases/jin_2014_density_versions_plunnecke_inequality/lemma_1|Lemma 1]] converts a minimal forward density on an interval into sumset growth. Choosing the last endpoint that attains each successive minimum partitions every initial interval into blocks with increasing densities. Summing their growth estimates proves the theorem at every cutoff.

The external input is [[additive_bases/jin_2014_density_versions_plunnecke_inequality/theorem_3|Theorem 3]], a truncated additive-graph form of Plünnecke's inequality. Jin states it on p. 3 and refers to proofs elsewhere on p. 4. Its graph proof is not in this manuscript and is not reproduced here. Jin's argument avoids the impact function used in the treatments cited as Jin's references [10] and [9] (Plünnecke 1970 and Nathanson 1996).

Other densities. [[additive_bases/jin_2014_density_versions_plunnecke_inequality/theorem_4|Theorem 4]] proves a lower asymptotic density inequality. The corresponding [[additive_bases/jin_2014_density_versions_plunnecke_inequality/theorem_5|upper asymptotic version fails]], by a counterexample attributed to Jin's 2011 paper. [[additive_bases/jin_2014_density_versions_plunnecke_inequality/theorem_6|Theorem 6]] treats upper Banach density, and [[additive_bases/jin_2014_density_versions_plunnecke_inequality/theorem_7|Theorem 7]] treats lower Banach density using the upper Banach density of $hB$. These pages record precise statements, proof sketches or external pointers, and their remaining proof coverage. They are not complete rewritten proofs.

Source corrections and endpoints. In Lemma 1 the translated set is explicitly truncated to the interval before applying Theorem 3. The printed proof on p. 15 says $z=\min A_0$ where the chosen subset requires $z=\min A'$; the result page proves the needed bound for every nonempty $A'$. The density statements separate order one at zero density instead of assigning a value to $0^0$. Jin's reference [10] prints volume 234 for Plünnecke's 1970 paper; the correct volume is 243. Jin's introductory date 1937 for Erdős's theorem is not used here: the existing [[additive_bases/erdos_1936_arithmetical_density_sum_two_sequences_one/_index|Erdős source]] records the primary paper's 1935 date and its Er36c archive key.

Alternative proofs and later literature. A bounded search identified the following distinct sources. Their proofs remain outside this source's coverage.

• J. L. Malouf, “On a Theorem of Plünnecke Concerning the Sum of A Basis and A Set of Positive Density,” Journal of Number Theory 54 (1995), 12–22, DOI 10.1006/jnth.1995.1098. The publisher's abstract advertises another simplified proof of the same Schnirelmann bound. Its proof was not acquired or compared with Jin's.
• Jin's earlier “Plünnecke's theorem for asymptotic densities,” Transactions of the AMS 363 (2011), 5059–5070, DOI 10.1090/S0002-9947-2011-05533-9, uses nonstandard analysis. This is reference [8], which the manuscript cites under a different title, “Plünnecke's Theorem for other densities.” It also supplies the un-repeated proof of Theorem 5.
• Michael Björklund and Alexander Fish, “Plünnecke inequalities for countable abelian groups,” Journal für die reine und angewandte Mathematik 730 (2017), 199–224, DOI 10.1515/crelle-2014-0129, arXiv:1311.5372, develops related inequalities using measure-preserving actions and ergodic bases. This is a later extension of the density methods, not evidence that every theorem about these densities has been compiled here.
• Jin explicitly points on p. 7 to stronger specialized bounds in Ruzsa's 1988 paper on squares and primes and 1989 paper on powers of primes. Ge's 2022 paper on polynomial values extends this specialized line. The prime and cube corollaries recorded under Theorem 4 are therefore examples, without a claim to be current best bounds. The strengthened proofs require separate compilation.

Formalization. Mathlib documents related finite-set Plünnecke–Petridis and Plünnecke–Ruzsa inequalities. That documentation does not provide the exact truncated statement used here or Jin's density proof. No matching formalization was located in the bounded search, and no Lean build was run.

Bears on.

• #35: Theorem 2 implies the requested quantitative increment, as proved at the end of its result page.
• #37: by Theorem 2, every Schnirelmann basis (which contains 0) is an essential component, since $\alpha^{1-1/h}>\alpha$ for $0<\alpha<1$. This is the basis case of the notion the problem defines; it says nothing about lacunary sets, which the question is about.
• #38: Theorem 2 bounds the whole sumset $A+B$ when $B$ is a Schnirelmann basis. The problem asks about sets $B$ that are not bases and about a single shift $A\cup(A+b)$, and the theorem asserts neither.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.