Erdős discrepancy problem

Erdős discrepancy problem
Name	Erdős discrepancy problem
Field	Combinatorics, Number theory
Introduced	1932s
Proposed by	Paul Erdős
Solved	2015
Solver	Terence Tao

Erdős discrepancy problem

The Erdős discrepancy problem is a question in Paul Erdős's body of problems concerning sign sequences and their irregularities, asking whether every infinite sequence taking values ±1 has unbounded discrepancy along homogeneous arithmetic progressions; the problem links concepts from Combinatorics, Number theory, Harmonic analysis, Ergodic theory, and Theoretical computer science. It was posed by Paul Erdős and remained a central open problem discussed in seminars at institutions such as University of Chicago and Institute for Advanced Study before a full resolution by Terence Tao in 2015, building on partial results by researchers connected to groups like University of California, Los Angeles and projects at Microsoft Research.

Introduction

The problem arises from Erdős's investigations into irregularities of sign patterns studied alongside contemporaries like Paul Turán and Pál Turán's students; it asks about sequences of ±1 values indexed by the natural numbers and their cumulative sums along multiples of integers. Early related work by Vinogradov-era analysts and combinatorialists such as Erdős and Alfréd Rényi framed discrepancy questions that intersected with results from Ramsey theory, Szemerédi's theorem, and conjectures circulated at venues including Princeton University and Cambridge University seminars. Partial and finite-case investigations engaged mathematicians at institutions like Massachusetts Institute of Technology and Harvard University.

Statement of the problem

Given a sequence f: N → {+1, −1}, define the discrepancy for an integer n and stride d by the sum S(d,n) = Σ_{k=1}^n f(kd). The Erdős discrepancy problem asks whether for every infinite f taking values ±1, the quantity sup_{d,n} |S(d,n)| is infinite. Erdős formulated this alongside questions about multiplicative functions studied by Dirichlet and G.H. Hardy, and the statement connects to classical objects like Möbius function and conjectures in Analytic number theory. The finite variant asks for bounds on discrepancy for sequences of fixed length N, a setting explored in computational projects at labs such as Google and research groups at University of Cambridge.

Historical context and partial results

Erdős posed the problem in the context of discrepancy theory developed concurrently by Donald Knuth-era computer scientists and combinatorialists like József Beck and Jeremy Spencer. Early partial results included constructions showing arbitrarily large discrepancy for certain structured sequences, with notable contributions by Leo Moser and combinatorialists at University of Illinois Urbana-Champaign. Computational searches and SAT-solver approaches were applied by teams including researchers at University of Minnesota and industrial labs; they produced finite lower bounds and counterexamples to restricted formulations. Work by Ben Green and Terence Tao on arithmetic progressions provided tools that influenced approaches, and results by Noga Alon and János Komlós on combinatorial discrepancy offered conceptual frameworks. Prior to the full proof, there were also lower-bound results motivated by techniques from Probabilistic method pioneers like Paul Erdős himself and practitioners at Bell Labs.

Proof of the finite case and Tao's resolution

The finite case—establishing deterministic bounds for sequences up to a given length—was extensively explored using computational and combinatorial methods by collaborators including those at University of Oxford and projects affiliated with CNRS. The complete resolution came with Terence Tao's 2015 proof, announced from University of California, Los Angeles and circulated via archives and seminars at Institute for Advanced Study. Tao's proof combined Fourier-analytic ideas linked to work by Norbert Wiener and Antoni Zygmund with multiplicative number theory traditions associated to Dirichlet and John von Neumann-era harmonic analysis. The argument reduced the problem to cases involving multiplicative functions and then employed techniques resonant with contributions from Elliott-type conjectures and work by H. Halberstam's school. Tao demonstrated that any ±1 sequence must yield unbounded discrepancy, completing the conjecture posed by Paul Erdős decades earlier.

Methods and techniques

Key methods in proofs and partial results include Fourier analysis on the integers, pretentious multiplicative function theory developed in schools following Harold Davenport and Andrew Granville, and ergodic-theoretic perspectives influenced by Furstenberg's correspondence principle. Combinatorial discrepancy methods from Beck-Fiala theorem contexts and probabilistic constructions by Erdős-style random sequences informed lower bounds. Computational techniques leveraged SAT solvers and exhaustive search strategies pioneered in algorithmic research at Carnegie Mellon University and Stanford University. Tao's synthesis drew on multilinear forms and decomposition theorems whose antecedents trace to work by Gowers and analysts at University of Cambridge.

Related problems and generalizations

The problem connects to the Möbius function randomness conjectures and the Chowla conjecture in analytic number theory, to discrepancy problems in higher dimensions studied by József Beck and Ronald Graham, and to sequence regularity questions considered by Kurt Mahler and Carl Ludwig Siegel. Generalizations include replacing ±1 values by bounded real sequences, studying discrepancy over polynomial progressions as in research influenced by Bourgain and Terence Tao's collaborators, and exploring algorithmic complexity versions considered in computational complexity seminars at Massachusetts Institute of Technology and Princeton University. Connections also arise with equidistribution topics relating to Weyl and uniform distribution theory developed across European schools.

Applications and open questions

While primarily a theoretical result, implications touch on pseudorandomness in constructions used by researchers at IBM Research and the design of sequences in coding theory circles at Bell Labs and Nokia Bell Labs. Open questions include quantitative bounds on growth rates of discrepancy, finer classifications of sequences attaining near-minimal discrepancy, and extensions to multidimensional index sets studied by teams at École Normale Supérieure and Technion. Further research continues in centers such as Princeton University, University of Cambridge, and Institute for Advanced Study, exploring refinements related to multiplicative functions and ergodic methods.

Category:Combinatorics