Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.

Updated Oct 8

Discrepancy

astashkin_2024_random_unconditional_convergence_rademacher_chaos_discrepancy/: Proves two-sided estimates for the discrepancy of edge-weighted graphs and homogeneous hypergraphs in terms of the weights, extending the Erdős and Spencer order n to the 3/2 for the complete graph, through random unconditional convergence of Rademacher chaos in L-infinity.

champagne_2024_well_distribution_modulo_one_primes/: Constructs an irrational number whose products with the primes fail to be well-distributed modulo one.

conlon_2011_large_almost_monochromatic_subsets_hypergraphs/: Shows every l-coloring of the triples of an N-set has a subset of size c(l, epsilon) sqrt(log N) with all but an epsilon fraction of its triples one color.

erdos_1949_uniform_distribution_modulo_1_lacunary_sequences/: Gives an almost-everywhere discrepancy bound for lacunary sequences, which the authors call sharper than all known results for sequences of the shape theta times lambda_n.

erdos_1963_ramsey_es_van_der_waerden_tetelevel/: Bounds how unbalanced a two-coloring must be on some complete subgraph or on some arithmetic progression, giving logarithmic and n^{3/2} bounds.

erdos_1964_problems_results_diophantine_approximations/: A survey of Erdos's problems on discrepancy, uniform and well distribution, and metric diophantine approximation, with many open questions.

erdos_1971_imbalances_colorations/: Determines the largest unavoidable imbalance in a plus-minus coloring of the k-subsets of n points to be of order n to the power (k+1)/2.

kesten_1966_bounded_remainder/: Kesten’s bounded-discrepancy length criterion, consolidated primary source and proof obligations.

khintchine_1923_ein_satz_uber_kettenbruche_mit/: Proves the sum of the first n partial quotients is almost always o(n^{1+e}), applies this to almost-everywhere discrepancy bounds for the multiples kx, and poses Khintchine's question on measurable sets (Problem 994).

marzo_2021_discrepancy_minimal_riesz_energy_points/: Upper bounds are proved for the spherical cap discrepancy of minimizers of the Riesz s-energy on the d-sphere, improving earlier bounds in several ranges of s.

openai_2026_euclidean_steinitz_bergstrom_theorem/: A 29-page release manuscript claiming that every finite sequence in the Euclidean unit ball of $\mathbb R^d$ has one signing with all prefix sums of norm at most $C\sqrt d$ , uniformly in length, so $S_2(d)=\Theta(\sqrt d)$ ; Dirichlet-energy body plus Gaussian correlation; the upper bounds are formally verified here, the simplex lower bound, the sharpness examples and the corollaries are not; background for Problem 178.

roth_1964_remark_concerning_integer_sequences/: Shows every set of integers up to N of density eta has, at some m <= N and some modulus q <= N^{1/2}, mean squared discrepancy over the residue classes

eta(1-eta)N^{1/2}.

schmidt_1972_irregularities_distribution/: Shows that for every sequence in the unit interval the set of points alpha at which the discrepancy of [0, alpha) stays bounded is at most countable.

tao_2016_erdos_discrepancy_problem/: Proves that every sequence of plus and minus ones has unbounded sums along homogeneous arithmetic progressions.

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Linked from (15)

library discrepancy/astashkin_2024_random_unconditional_convergence_rademacher_chaos_discrepancy discrepancy/champagne_2024_well_distribution_modulo_one_primes discrepancy/conlon_2011_large_almost_monochromatic_subsets_hypergraphs discrepancy/erdos_1949_uniform_distribution_modulo_1_lacunary_sequences discrepancy/erdos_1963_ramsey_es_van_der_waerden_tetelevel discrepancy/erdos_1964_problems_results_diophantine_approximations discrepancy/erdos_1971_imbalances_colorations discrepancy/kesten_1966_bounded_remainder discrepancy/khintchine_1923_ein_satz_uber_kettenbruche_mit discrepancy/marzo_2021_discrepancy_minimal_riesz_energy_points discrepancy/openai_2026_euclidean_steinitz_bergstrom_theorem discrepancy/roth_1964_remark_concerning_integer_sequences discrepancy/schmidt_1972_irregularities_distribution discrepancy/tao_2016_erdos_discrepancy_problem

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