Littlewood conjecture

The open conjecture that every pair of real numbers admits arbitrarily strong simultaneous rational approximation with a common denominator in a multiplicative sense.

Version

v1 · 2026-09-08 · History

Domain-specific #

5365

Origin domain

diophantine approximation

Subdomain

diophantine approximation

Core Idea

The liminf statement is trivial when either number is rational or not badly approximable; any counterexample lies in a thin exceptional set and the conjecture remains unresolved. Multiples of two reals are compared with their nearest integers, and the conjecture asserts that the denominator times the product of both errors becomes arbitrarily small. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.

Scope of Application

Littlewood conjecture belongs to diophantine approximation and is useful where the analyst can specify the typed diophantine approximation carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, then evaluate the real pair, nearest-integer norm, positive integer denominator, exact liminf expression, known special cases and exceptional-set results and open status are explicit. The scope is broad within that domain but bounded by the need for the real pair, nearest-integer norm, positive integer denominator, exact liminf expression, known special cases and exceptional-set results and open status are explicit. The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.

Clarity

The abstraction clarifies a crowded vocabulary by making the real pair, nearest-integer norm, positive integer denominator, exact liminf expression, known special cases and exceptional-set results and open status are explicit the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test. A bare label is insufficient because the name Littlewood conjecture can be used for a formal identity, an implementation, or a neighboring result unless carrier and convention are stated.

Manages Complexity

Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Littlewood conjecture. Littlewood conjecture compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.

Abstract Reasoning

1. Identify the carrier. State what the elements, states, objects, or observations are: the typed diophantine approximation carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express the real pair, nearest-integer norm, positive integer denominator, exact liminf expression, known special cases and exceptional-set results and open status are explicit independently of one notation or implementation.

Knowledge Transfer

Knowledge transfers strongly among subfields of diophantine approximation because they reuse the typed diophantine approximation carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, Multiples of two reals are compared with their nearest integers, and the conjecture asserts that the denominator times the product of both errors becomes arbitrarily small., and type the carrier, state every parameter and convention in the definition, test that the real pair, nearest-integer norm, positive integer denominator, exact liminf expression, known special cases and exceptional-set results and open status are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions

Current abstraction Littlewood conjecture Domain-specific

Parents (1) — more general patterns this builds on

• Littlewood conjecture is a kind of Constraint Prime

  The proposed strict upward parent is prime:constraint.

Hierarchy path (1) — routes to 1 parentless root

• Littlewood conjecture → Constraint

Neighborhood in Abstraction Space

Littlewood conjecture sits in a moderately populated region (49^th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Algebraic Number Theory & Reciprocity (28 abstractions)

Nearest neighbors

• Diophantine quintuple — 0.90
• Irrationality measure — 0.89
• Cannonball problem — 0.88
• Miller–Rabin primality test — 0.88
• Faulhaber's formula — 0.88

Computed from structural-signature embeddings · 2026-09-08