The Gaussian moat problem

Walks through Gaussian primes with bounded steps. The arithmetic of split primes and the reduction of the uniform component bound to a finite periodic sieve; Shannon entropy, Pinsker's inequality, short lists of candidates and averages over random coordinates; Hoeffding's inequality and the isoperimetry of the discrete cube; many differences of a walk stretch and their separation by randomly signed products of Gaussian primes; entropy enrichment by sampling differences; coverage transferred backwards with shared continuations; a common sampling schedule; and the information cost of avoiding zero classes. Together they prove that for every step bound the connected components of the Gaussian-prime graph have bounded size, so that one cannot walk to infinity on Gaussian primes with bounded steps.

Eight lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.

1. Walks through Gaussian primes
2. Entropy of finite random variables
3. Concentration and the discrete cube
4. Differences of a walk and separating products
5. Entropy enrichment
6. Coverage from shared continuations
7. A common sampling schedule
8. Zero avoidance and the finite sieve

Theorems and assumptions · Reader and editable sources · Course record

Prerequisites

• Lattices, Minkowski's theorem and the Minkowski embedding · Number fields
• The prime number theorem for arithmetic progressions · Dirichlet L-functions and primes in progressions
• Counting primes by elementary means: Chebyshev and Mertens · The Riemann zeta function
• Euclidean domains and unique factorization (core course Abstract Algebra II), finite probability (core course Probability), and Brouwer's fixed-point theorem (core course Algebraic Topology, Lecture 30).