Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.
Updated
Analysis
anderson_1967_example_dimension_theory/: Constructs, for each n, a subset K of Euclidean n-space whose finite powers and countable power all have topological dimension n minus one, the same as K itself.
armentano_et_al_2025_characterization_logarithmic_fekete_critical_configurations_at_most_six_points_all_dimensions/: Classifies logarithmic Fekete critical configurations with at most six points on unit spheres in every relevant dimension by a finite Gram-matrix computation, with a clear but restricted connection to E1045.
atkinson_1961_problem_erdos_szekeres/: Sharpens the Erdos-Szekeres upper bound on the least possible maximum over the unit circle of a product of factors 1 - z^{a_k}: its logarithm is at most about the square root of n times log n.
atkinson_1961_sums_powers_complex_numbers/: Proves that the largest modulus of the first n power sums of complex numbers with z_1 = 1 and all moduli at most one exceeds 1/6.
bedert_2025_polynomial_bounds_chowla_cosine_problem/: Establishes the first polynomial bound for Chowla's cosine problem, showing any n-term cosine sum over positive integers dips below -n^{1/5-o(1)}.
belov_1996_estimate_free_term_nonnegative_trigonometric_polynomial/: Bounds the least constant term of a nonnegative cosine polynomial with nonincreasing nonnegative integer coefficients summing to n between (log n)^2/log log n and (log n)^3, giving log f(n) << (log n)^4 for the Erdos-Szekeres product.
biro_1994_problem_turan_concerning_sums_powers_complex/: Proves by Newton--Girard identities and a planar geometric dichotomy that the first n power sums have maximum modulus strictly greater than one half when one of the complex numbers is one.
biro_2000_improved_estimate_power_sum_problem_turan/: Proves that some effectively computable absolute constant q greater than one half bounds from below the largest modulus among the first n power sums of any complex numbers with z_1 equal to one, improving the one-half bound of Biró 1994.
biro_2000_upper_estimate_turan_pure_power_sum_problem/: Proves that the minimum over normalized complex n-tuples of the largest of the first n power sums has limit superior strictly less than one, with the explicit bound five sixths for large n and Harcos's computed 0.69368.
bonami_revesz_2007_integral_concentration_idempotent_trigonometric_polynomials_gaps/: Proves L^p concentration results for idempotent trigonometric polynomials, with and without large gaps; it neither answers E1150 affirmatively nor constructs ultraflat Littlewood polynomials.
bonami_revesz_saffari_2007_problemes_ouverts_theorie_analytique_polynomes_analyse_harmonique/: A theorem-indexed source review read from a complete Markdown transcription of the collection.
borichev_et_al_2017_spectra_stationary_processes_z/: Proves finite-valued stationary processes with a spectral gap are periodic, and records the exact limits of that rigidity method for E1150.
bourgain_1986_sur_le_minimum_d_une_somme/: Proves that a cosine sum with 0-1 Fourier coefficients and N frequencies has negative part of sup-norm at least super-logarithmic in N.
bourgain_2018_paper_erdos_szekeres/: Revisits the Erdős–Szekeres product problem: builds dense sets of n exponents whose maximum modulus is at most exp of a constant times root n times root log n times log log n, shows almost full sets force exponential growth, and bounds the product below through dissociated subsets.
brauchart_2024_complete_minimal_logarithmic_energy_asymptotics_points_compact_interval_consequence_discriminant_ja/: Gives the complete Fekete logarithmic-energy expansion on an interval and places its capacity term and finite-size corrections beside Pommerenke's general convex-support bounds.
cabrelli_lacey_molter_pipher_2005_journe_lemma_variations/: Variants of Journé's covering lemma for dyadic rectangles in two and more parameters, bounding embeddedness-weighted sums of rectangle areas by the measure of their shadow, with large or (1+delta)-small enlargements.
cambie_2026_maximum_product_distances_diameter_point_sets/: Constrains the structure of point sets maximizing the product of pairwise distances and beats the regular polygon for even orders.
carnielli_2011_adjusting_conjecture_erdos/: Disproves Erdos's reverse Littlewood-Offord conjecture in every dimension above one and proposes a corrected dimension-dependent version.
chojecki_2026_note_erdos_path_problem_transcendental_entire/: Answers two of Erdős's questions on paths to infinity for entire functions, and shows no fixed power of the maximum modulus can be forced.
clement_steinerberger_2025_balanced_stick_breaking/: Shows that for the golden-ratio Kronecker sequence and the base-2 van der Corput sequence the ratio of the largest to the smallest sum of r ≥ 2 consecutive gaps stays below 1 + c log r/r for all large n, so the third de Bruijn–Erdős constant satisfies μ_r ≤ 1 + c log r/r; an unrefereed preprint that bounds the growth in the third part of Problem 1221.
csaki_2005_frequently_visited_sets_random_walks/: Shows that the most visited translate of a finite set by a symmetric transient walk on Z^d with finite second moments is occupied about -log n/log(1 - 1/Lambda_A) times, Lambda_A the top eigenvalue of its Green matrix, with exact two-site and unit-sphere laws and a Brownian invariance principle.
danzer_pommerenke_1967_ber_die_diskriminante_von_mengen_gegebenen_durchmessers/: Disproves the regular-polygon conjecture for the largest product of distances among k planar points of diameter at most two at every even k from four on, finds the exact maximum for two, three and four points, and bounds it above by k to the k times exp of fifteen k to the six sevenths for large k.
debruijn_1951_functions_whose_differences_belong_given_class/: Proves a function whose every difference is continuous splits into a continuous plus an additive function, and finds which classes share this property.
debruijn_1966_almost_additive_functions/: Shows a function satisfying the additivity equation for almost all pairs of reals agrees almost everywhere with a genuinely additive function.
debruijn_erdos_1949_sequences_points_circle/: De Bruijn and Erdős's 1949 note on the largest and smallest sums of r consecutive gaps of a sequence on the circle: the exact r = 1 constants 1/log 2, 1/log 4 and 2, the bounds 1/log(1+1/r), (r/(r+1))/log(1+1/r) and 1 + 1/r for general r, and the closing conjecture behind Problem 1221.
dembo_2004_cover_times_brownian_motion_random_walks/: Gives sharp torus cover times and the limiting law for covering a planar disc.
dembo_2007_how_large_disc_covered_random_walk/: Distinguishes the origin-centered radius law from the much larger movable disc.
downarowicz_lacroix_1998_merit_factors_morse_sequences/: Relates unbounded binary merit factors to Morse spectral measures and gives a conditional dynamical route to the E1150 gap.
dvoretzky_1959_divergence_random_power_series/: Gives a condition on coefficient size under which almost all randomly signed power series diverge at every point of the unit circle.
edgar_2001_hausdorff_dimension_analytic_sets_transcendence/: Shows that a proper real closed subfield of the reals that is an analytic set has Hausdorff dimension zero, since analytic sets of positive dimension contain a transcendence base for the reals.
edgar_2003_borel_subrings_reals/: Proves that a Borel or analytic subring of the reals either has Hausdorff dimension zero, in the strong sense that all its finite powers do, or is the whole real line, with analogs for subrings of the complex and p-adic numbers.
erdos_1945_lemma_littlewood_offord/: Proves sharp real signed-sum bounds, the complex projection estimate, and distinct shadow and Menger proofs of central-level family bounds.
erdos_1949_strong_law_large_numbers/: Constructs a periodic function and a lacunary sequence for which averages of its values diverge, and sharpens the Kac-Salem-Zygmund condition.
erdos_1950_distribution_roots_polynomials/: Proves a quantitative bound showing the arguments of a polynomial's roots are equidistributed up to an error controlled by its coefficient sizes.
erdos_1954_integral_functions_gap_power_series/: Gives sharp gap conditions under which an entire lacunary series has minimum modulus asymptotic to its maximum modulus along a sequence of radii.
erdos_1955_polynomials_whose_zeros_lie_unit_circle/: Constructs a polynomial with all zeros on the unit circle whose modulus is below one and above one somewhere on every radius of the disc.
erdos_1959_product/: Shows the least possible maximum modulus of a product of n terms of the form one minus z to a power grows subexponentially.
erdos_1960_problems_concerning_structure_random_walk_paths/: Analyses returns, distance growth and point multiplicities for lattice random walks, and finds the largest multiplicity in dimension three and above.
erdos_1964_arithmetical_tauberian_theorems/: Gives a Tauberian equivalence for nondecreasing real sequences with finite reciprocal sum, and separately asks about zeros for distinct integers.
erdos_1966_additive_gruppen_mit_vorgegebener_hausdorffscher_dimension/: Constructs additive groups of real numbers of every prescribed Hausdorff dimension between zero and one, using Cantor-series digit conditions.
erdos_1973_remark_polynomials_transfinite_diameter/: Shows the sublevel set where a monic polynomial has modulus below one contains a disc of radius depending only on the transfinite diameter, when the zeros lie in a connected compact set of transfinite diameter below one.
erdos_1976_extremal_problems_polynomials/: Records Erdős's 1976 formulations of the diameter-constrained distance-product problem and the Littlewood-polynomial maximum-modulus conjecture.
erdos_1981_correction_misprints_our_paper_almost_everywhere/: A one-page errata table correcting misprints in the authors' paper on almost everywhere divergence of Lagrange interpolation polynomials.
ghosh_2024_number_components_polynomial_lemniscates_problem_erdos/: Shows the maximal number of lemniscate components is eventually below a fixed fraction of n when the capacity is under one, and equals n for all large n when the capacity exceeds one and the set is connected or has a regular equilibrium measure.
glucksam_2024_approximate_solution_erdos_maximum_modulus_points/: Constructs an entire function whose count of arcs of approximate maximum modulus on each circle tends to infinity, as evidence on Erdos's question.
goldberg_1978_counting_functions_sequences_points_entire_functions/: Constructs an entire function whose unintegrated counting functions of a-points have ratio with upper limit infinity and lower limit zero, answering an Erdos question.
goldberg_1979_asymptotic_curves_entire_functions_finite_order/: Disproves the Hayman-Erdos conjecture that an entire function of finite order has an asymptotic curve of length l(r) = O(r).
goldberg_1979_sets_which_modulus_entire_function_has/: Proves Hayman's conjectured growth bound for entire functions whose superlevel set has finite area and shows it is sharp, answering Erdos negatively.
goodman_1966_convexity_level_curves_polynomial/: Gives two quartic counterexamples to Grunsky's question whether the m components of a sublevel set of a polynomial with m distinct roots must be convex, the second with four simple roots, and records open questions on starlikeness.
grow_whicher_1984_finite_unions_quasi_independent_sets/: Grow and Whicher give a finite 15-element obstruction to the exact two-class analogue of Horn's theorem and reduce the infinite covering question over the integers to a uniform finite covering question.
hanninen_2018_sparse_carleson_coefficients_general_sets/: Hänninen's theorem that, for a locally finite Borel measure on R^d with no point masses and any countable collection of Borel sets, Carleson and sparse coefficients coincide, with the same constant.
hao_2024_favorite_sites_simple_random_walk_two/: Proves the planar simple-random-walk favorite-count limit superior three and the sharp iterated-logarithm limit-superior law in dimensions at least three.
hare_yang_2018_sidon_sets_proportionally_sidon_small_constants/: Strengthens Pisier's local extraction theorem in torsion-free groups by obtaining large bounded-degree-independent subsets and Sidon constants arbitrarily close to one.
harrison_ramsey_1996_partitioning_sidon_sets_quasi_independent_sets/: Gives random Sidon sets with finite bounded-relation-independent partitions and finite-determination principles for the unresolved general partition problem over the integers.
he_2024_reverse_littlewood_offord_problem_erdos/: Gives an elementary pairing proof that a random signed sum of planar unit vectors lies in a ball of radius root two with probability at least c over n.
he_2026_generalizing_clunie_hayman_construction_erdos_maximum/: Improves the lower bound for Erdos's maximum-term constant from 4/7 to 0.58507 via a two-parameter Clunie-Hayman construction.
ho_2026_counterexamples_lacunary_dilates_via_dyadic_spike/: Builds functions on the circle and dyadic lacunary sequences along which the averages of f(n_j x) diverge almost everywhere, answering Erdos Problem 996 and the example question of Problem 995 in the negative.
hollom_2025_double_jump_phase_transition_reverse_littlewood_offord/: Disproves the odd-n unit-radius form of Erdős's reverse Littlewood-Offord conjecture with an O(n^{-3/2}) construction, proves the order-1/n bound at every radius above one and an exponential bound at radius one, and compares orthogonal, simplicial and mixed minimizers at radius root d.
hollom_sorkin_2025_reverse_littlewood_offord_parity_conditions/: Constructs odd-n planar unit vectors whose signed sum lies in the closed unit disk with probability exactly 2^{-floor(n/2)}, and proves that for n of parity opposite to d some signed sum of unit vectors in d-space has norm at most root of d minus epsilon, with epsilon = 2^{-100} d^{-80}.
huang_2025_many_lemniscates_large_diameter/: Constructs monic polynomials whose sublevel set has arbitrarily many components of diameter near 4, refuting a bounded-count question of Erdos.
janson_1998_new_versions_suen_correlation_inequality/: Janson's strengthened forms of Suen's correlation inequality for dependent indicator variables.
jin_2025_small_eigenvalues_large_cuts_chowla_s/: Proves graphs with small least eigenvalue contain large cliques, giving the first polynomial bound for Chowla's cosine problem.
jung_2024_fifty_years_erdos_similarity_conjecture/: Surveys progress on the Erdos similarity conjecture and its bi-Lipschitz, topological and large-scale variants after fifty years.
jurkat_1965_cauchy_functional_equation/: Gives an independent conull-sumset proof that an almost-everywhere solution of Cauchy's equation has a unique everywhere additive correction.
keleti_1998_difference_functions_periodic_measurable_functions/: Records essential-continuity difference results and the classification of essentially-continuous-difference shift subgroups as the finite subgroups.
kleitman_1965_lemma_littlewood_offord_distribution_certain_sums/: Proves the sharp complex signed-sum bound through symmetric chains and two-color Sperner, while preserving the later geometric proof limits.
konyagin_1981_littlewood_problem/: Proves the Littlewood conjecture that the L1 norm of a sum of M exponentials with integer frequencies, not necessarily distinct, is at least a constant times log M.
korsky_2026_improved_lower_bound_debruijn_erdos_consecutive_gap_problem/: Proves that for every r at least 2 and every sequence of distinct points on the circle the upper limit of the ratio of the largest to the smallest sum of r consecutive gaps is at least 1 + r/(r^2 − 1), improving the 1949 bound 1 + 1/r for each fixed r; an unrefereed note.
korsky_2026_resolution_debruijn_erdos_consecutive_gap_problem/: Claims all three parts of the 1949 de Bruijn–Erdős conjecture in mean-normalized form for sequences of distinct points, with the two one-sided excesses at least a constant times root log r and the ratio at least 1 + log r/(100 r); an unrefereed, AI-assisted preprint registered as a proof claim for Problem 1221, with no acceptance evidence found.
laczkovich_1980_functions_measurable_differences/: Records the measurable-summand weak difference theorem and its distinction from the continuous-summand formulation of Problem 908.
laczkovich_1984_kemperman_s_inequality/: Gives the full finite-difference and continued-fraction proof that Kemperman’s inequality forces nondecreasing real functions.
lefevre_et_al_2009_thin_sets_integers_harmonic_analysis_p_stable_random_fourier_series/: Relates Sidon, Rider, and quasi-independent extraction estimates to the proportional-dissociation formulation of E0774.
lewko_2026_sidon_decomposition_problem_abelian_groups_bounded_torsion/: Lewko proves that Sidon sets in bounded-torsion abelian groups are finite unions of quasi-independent sets, using relation-space dimension bounds and a Rado--Horn support partition; the method does not cover the integers.
michelen_2025_convergent_points_random_power_series_unit/: Proves that a random sign power series with coefficients of size o(1/sqrt n) almost surely converges on a set of Hausdorff dimension one on the unit circle.
murai_1983_deficiency_entire_functions_fejer_gaps/: Proves an entire function whose exponent set has convergent reciprocal sum has no finite deficient value, and that this fails under Fabry gaps.
openai_2026_dyadic_case_erdos_similarity_conjecture/: A sixteen-page manuscript of the OpenAI mathematics release claiming the dyadic case of the Erdős similarity conjecture (Problem 120): for every a compact of measure above contains no affine copy , of either sign, built from random periodic hitting sets routed through a finite tree and rescaled dyadically.
openai_2026_geometric_case_erdos_similarity_conjecture/: A thirteen-page release manuscript claiming the geometric-progression case of the Erdős similarity conjecture: for each fixed ratio q in (0,1) and each eta, a compact subset of [0,1] of measure above 1-eta that contains no nontrivial affine copy of {q^n : n >= 1}, built by a random routing construction on a finite ordered tree of dyadic grids; bears on Problem 120.
oriike_2026_negative_answer_universal_function_version_erdos/: Shows no fixed comparison function forces every transcendental entire function to have a path where its modulus outgrows that function of the maximum modulus.
pendyala_2026_shortest_paths_polynomial_lemniscate_sublevel_sets/: Claims bounds of order root log n from below and pi times n from above for the extremal escape-path length in a lemniscate problem of Erdős.
pisier_1983_arithmetic_characterizations_sidon_sets/: Pisier characterizes Sidon sets by the existence of linearly sized quasi-independent subsets in every finite subset, which, he says, in some sense reduces the finite-union question for Sidon sets to a combinatorial question whose case of sets of positive integers is E0774.
pommerenke_1959_some_problems_erdos_herzog_piranian/: Pommerenke's 1959 answers to questions of Erdős, Herzog and Piranian on the lemniscate |f(z)| = 1 of a monic polynomial: a lemniscate lying within distance 2 of one of its own support points, and, when the interior E is connected, length at least 2π, containment in the disc of radius 2 about the centroid of the zeros, bounds on the diameter and width, and a width above 2.18 that refutes the conjectured bound 2.
pommerenke_1961_metric_properties_complex_polynomials/: Answers to 1958 problems of Erdős, Herzog and Piranian on the set where a monic polynomial has modulus at most one: unboundedly many components of diameter near 4 (Theorem 1), the z^n - r^n example (p. 98), a disk of radius (2e)^{-1} n^{-2} (Theorem 4), every projection above 2.386 (p. 103), length below 74 n^2 (Theorem 9), a connected set inside the disk of radius 2 about the centroid (Theorem 10), a nonconvex component (Theorem 14) and the discriminant bound 2^{4(n-1)} n^n (Theorem 16).
ramsey_graham_2006_permutation_extension_planar_quasi_independent_subsets_roots_unity/: Characterizes the permutations of the n-th roots of unity that preserve their quasi-independent or independent subsets in the plane, and bounds the largest quasi-independent subset as a prime factor of n is enlarged.
ramsey_graham_2006_planar_sidonicity_quasi_independence_multiplicative_subgroups_roots_unity/: Ramsey and Graham study which sets of roots of unity are Sidon, independent or quasi-independent in the additive group of the complex numbers, with a square-free coset reduction, a local form of Pisier's criterion, and exact values of the largest quasi-independent subset of the n-th roots of unity for several families of n.
rudin_1958_connected_subset_plane/: Under the continuum hypothesis, constructs a connected planar set every non-degenerate connected subset of which has countable complement in it.
saff_sheil_small_1974_coefficient_integral_mean_estimates_restricted_zeros/: Proves sharp integral-mean bounds for algebraic polynomials whose zeros lie on the unit circle and for two-sided trigonometric polynomials whose zeros are all real.
sothanaphan_2025_improved_lower_bound_erdos_problem_concerning/: Constructs, for every large even n, planar point sets of diameter two whose squared product of pairwise distances exceeds 1.037 n^n, beating the regular n-gon by a constant factor.
steinhaus_1920_sur_les_distances_des_points_dans/: Proves a set of positive measure contains infinitely many points with pairwise rational distances and that its distance set contains an interval.
toppila_1976_counting_function_values_meromorphic_function/: Constructs a meromorphic function whose value counting functions have unbounded ratio for every pair of distinct values, answering a question of Erdős.
toth_2001_three_favorite_sites_simple_random_walk/: Proves that simple symmetric random walk on the integers almost surely has four or more favorite (most visited) sites at only finitely many times, by bounding the expected number of steps onto one of exactly four favorites, and leaves the case of three favorites open.
wang_2026_proposed_solution_erdos_problem_1002/: Claims that the normalized discrepancy sum of a random rotation converges to a centered Cauchy law with scale one over two pi.
wu_1985_length_paths_subharmonic_functions/: Sharpens length estimates for paths along which a subharmonic function grows, improving the Lewis-Rossi-Weitsman strengthening of Hall's lemma.
yip_2025_problem_erdos_ingham/: Represents arbitrary complex values by absolutely convergent sums of reciprocal powers and gives a negative result for the Erdős--Ingham infinite-sequence question, with an authored p. 2 refinement supplement.
This folder holds sources whose primary subject is Analysis.
Sources with other primary subjects
Explicit links to this subject's problems support these cross-references.
- kolountzakis_1996_density_b_h_g_sequences_minimum
- erdos_1964_problems_results_diophantine_approximations
- alexeev_2026_short_proofs_combinatorics_probability_number_theory
- erdos_1978_set_theoretic
- erdos_1982_my_favourite_problems_which_recently_have
- feng_2026_semi_autonomous_mathematics_discovery_gemini_case
- erdos_1961_unsolved_problems
- erdos_1965_recent_advances_current_problems_number_theory
- schmidt_1969_disproof_conjectures_diophantine_approximations
- various_1999_some_pauls_favorite_problems
- erdos_1958_metric_properties_polynomials
- hayman_lingham_2018_research_problems_function_theory
- krishnapur_2025_area_polynomial_lemniscates
- wang_2026_proposed_complete_solution_erdos_problem_1038