Large Set (Combinatorics)¶
A large set of positive integers has a divergent sum of reciprocals, a criterion finer than merely being infinite or having positive density.
Core Idea¶
A positive-integer set A is large when Σ_{a∈A}1/a diverges, and small when it converges. Even numbers and primes are large; squares are small. Infinitude and natural density are different tests: the primes have zero natural density yet divergent reciprocal sum.[^ref-dcb4c894387a]
Scope of Application¶
For positive integer exponents with constant 1 included, Müntz–Szász says the span of 1 and x^a is dense in C[0,1] exactly for large exponent sets. Bloom–Sisask proves every large set has a three-term progression. The arbitrary-length statement for every large set remains Erdős's conjecture, though Green–Tao proved arbitrary lengths for the particular large set of primes.[ref-fc70ff3ff56b][ref-c7d99cacd0ab][^ref-1622a83cd5a0]
Clarity¶
Σ1/(2n)=(1/2)Σ1/n diverges; Σ1/n² converges. Finite changes do not alter the classification. “Large” here is not a synonym for physically large, positive-density or merely infinite.
Manages Complexity¶
One weighted-series threshold supports comparison and monotonicity across distinct integer families, but downstream conclusions require separately stated theorems and hypotheses.
Abstract Reasoning¶
For even exponent set {2,4,6,...}, reciprocal divergence and a constant term give density of span{1,x²,x⁴,...} by the stated Müntz theorem. For square exponents {1,4,9,...}, reciprocal convergence blocks that conclusion. For primes, divergence and zero density coexist; the general three-term theorem and prime-specific Green–Tao result have different scopes.[ref-fc70ff3ff56b][ref-dcb4c894387a][ref-c7d99cacd0ab][ref-1622a83cd5a0]
Knowledge Transfer¶
The exact reciprocal test transfers between approximation and additive combinatorics; their conclusions do not. Do not promote the open arbitrary-length conjecture to a theorem or apply the integer-exponent form indiscriminately to arbitrary real exponents. The live Set and Membership prime is the strict genus; reciprocal divergence is the child's differentia.
[^ref-dcb4c894387a]: Tao, 254A number-theory notes. [^ref-fc70ff3ff56b]: Borwein and Erdélyi, “The Full Müntz Theorem in C[0, 1] and L1[0, 1]”, introduction, pp. 1–2. [^ref-c7d99cacd0ab]: Bloom–Sisask, three-term theorem. [^ref-1622a83cd5a0]: Green–Tao, prime progressions.
Relationships to Other Abstractions¶
Current abstraction Large Set (Combinatorics) Domain-specific
Parents (1) — more general patterns this builds on
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Large Set (Combinatorics) is a kind of Set and Membership Prime
A reciprocal-large set is a set of positive integers with divergent reciprocal sum.
Hierarchy path (1) — routes to 1 parentless root
- Large Set (Combinatorics) → Set and Membership
Neighborhood in Abstraction Space¶
Large Set (Combinatorics) sits in a sparse region of the domain-specific corpus (81st percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Number-Theoretic Properties & Tests (20 abstractions)
Nearest neighbors
- Yule–Simon Distribution — 0.83
- Average Order of an Arithmetic Function — 0.82
- Blackwell–Girshick Equation — 0.82
- Kolmogorov's Three-Series Theorem — 0.82
- Least Common Multiple — 0.82
Computed from structural-signature embeddings · 2026-10-08