Angular discrepancy bounds in terms of polynomial sparsity
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Can the degree in an Erdős–Turán angular-discrepancy bound for polynomial roots be replaced by the number ν of nonzero terms, giving a bound of order √(ν log M) in terms of Mahler measure M?
Why it matters and what counts as progress
Why it matters: A sparse bound of this form would control root angles through description size rather than degree.
What counts as progress: A universal square-root bound or sparse polynomials with controlled Mahler measure and angular discrepancy exceeding it, with multiplicities counted consistently.
How the problem developed
- 1950 · Theorem [1]
Paul Erdős and Pál Turán Proved a degree- and coefficient-sensitive bound for angular discrepancy of polynomial roots, establishing the classical baseline for the problem.
- 1972 · Partial result [2]
W. K. Hayman Developed angular value-distribution results for power series with gaps, a related line of work on sparse supports.
Alexeev and coauthors, with an internal OpenAI model Gave an explicit polynomial family refuting a proposed fewnomial angular-discrepancy bound; the degree-based Erdős–Turán estimate remains distinct.
AI contribution
Short proofs in combinatorics, probability and number theory II
There is no universal Erdős–Turán angular-discrepancy bound of order √(ν(f) log M(f)) with polynomial degree replaced by the number ν(f) of nonzero terms. For each N, a polynomial with ν=N+2 and M<3 has a positive real zero of multiplicity N+1, giving discrepancy at least N+1/2 on the stated short arc.
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