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

Updated Oct 8

Additive Bases and Sidon Sets

alexeev_2025_forbidden_sidon_subsets_perfect_difference_sets/: Disproves Erdos's prize conjecture that every finite Sidon set extends to a finite perfect difference set, with Lean-verified counterexamples.

alon_1985_application_graph_theory_additive_number_theory/: Shows every B_2^{(k)} sequence of n terms splits into at most c(k) n^{1/3} Sidon sets and contains a Sidon subset of at least c(k) n^{2/3} terms, and formulates Pisier's finite-union question for dissociated sets.

balasubramanian_2001_additive_complements_squares/: Improves the lower bound 4/pi on liminf b(N)/sqrt(N), b(N) the size of a minimal additive complement of the squares up to N, assuming that for a fixed delta in (0,1) and all large N some minimal complement lies in [0, delta N].

balogh_2021_upper_bound_size_sidon_sets/: Improves the classical upper bound for the largest Sidon set in the first n integers to n^(1/2) plus 0.998 times n^(1/4) for all large n, with analogous bounds for weak Sidon and t-thin Sidon sets.

barany_et_al_2023_lagrange_like_spectrum_perfect_additive_complements/: Proves that the set of limsup values of A(x)B(x)/x over perfect additive complements is closed, describes its discrete part below the first accumulation point, and locates an interval it contains and one it misses.

bennett_bohman_2013_note_random_greedy_independent_set_algorithm/: Proves the random greedy independent set process on a regular uniform hypergraph with small codegrees outputs at least order N(log N/D)^(1/(r-1)) vertices, a lower bound where E156 needs an upper bound.

bhalla_2026_regularly_thin_minimal_asymptotic_basis_order/: Constructs a minimal asymptotic basis of order two whose counting function equals C times the square root of x plus a bounded error.

borwein_et_al_2005_old_conjecture_erdos_turan_additive_bases/: Recasts the Erdős–Turán conjecture through generating functions and proves that if f has nonnegative integer coefficients and every coefficient b_n of f(z)^2 is positive, then the supremum of the b_n is at least 8.

bosio_2026_large_b_2_g_subsets_first/: Constructs B_2[g] subsets of the first n squares of size at least a constant times n^(2g/(2g+1)) (log n)^((2-2^g)/(2g+1)), for every fixed g.

carlet_2022_apn_functions_whose_graphs_are_maximal_sidon_sets/: Proves an APN graph is a maximal Sidon set in (F_2^n)^2 exactly when its triple sums cover the group, and asserts maximality for plateaued APN functions, which fails for n at most 2; a coverage model for E156.

carter_2025_diameter_finite_sidon_sets/: Improves the Erdos-Turan bound, showing a k-element Sidon set has diameter at least k^2 - 1.96365 k^{3/2} - O(k).

chen_2017_additive_complements_squares/: Shows every additive complement of the squares has an unbounded excess of representations, and that its n-th term falls below (pi^2/16)n^2 by more than 0.57 n^(1/2) log n infinitely often.

cilleruelo_1993_additive_completion_kth_powers/: Improves the lower bound on the size of a set that additively completes the kth powers up to N, with an explicit gamma-function constant.

cilleruelo_1995_b_2_g_sequences_whose_terms/: Shows the Erdos-Renyi bounded-representation sequence can be taken inside the squares, giving squares of numbers growing at most like k to the one plus epsilon.

cilleruelo_2000_upper_bound_b_2_2_sequences/: Proves that a set in [1,N] where no integer has more than two representations a + b with a <= b has at most sqrt(6N)+1 elements.

cilleruelo_2001_infinite_b_2_g_sequences/: Constructs infinite sequences in which each integer has a bounded number of representations as a sum of two terms, with a larger limit superior of A(x)/sqrt(x) than previously known.

cilleruelo_2002_upper_lower_bounds_finite_b_h/: Proves upper bounds for finite bounded-multiplicity sum sets that improve the counting bound, the best the paper says was known for multiplicity above one, and dense constructions for sums of two.

cilleruelo_2008_perfect_difference_sets_constructed_sidon_sets/: Builds perfect difference sets from dense Sidon sets, giving one with counting function A(x) >> x^(sqrt2-1+o(1)) and one with limsup A(x)/sqrt(x) >= 1/sqrt2.

cilleruelo_2010_generalization_theorem_erdos_renyi_sidon_sequences/: Gives two new proofs that bounded-multiplicity sum sequences can be almost as dense as the counting limit, with a much better multiplicity bound.

cilleruelo_2010_generalized_sidon_sets/: Determines the asymptotic size of the largest sets with bounded representation function in an interval and in cyclic groups as the bound grows.

cilleruelo_2010_probabilistic_constructions_b_2_g_sequences/: Constructs infinite sequences with at most g representations of each sum whose kth term is at most k to the power two plus one over g, up to a logarithmic factor.

cilleruelo_2011_concentration_points_two_three_dimensional_modular/: Bounds points of two- and three-variable modular hyperbolas in short boxes, showing both counts are subpolynomial for small enough boxes.

cilleruelo_2013_dense_sets_integers_prescribed_representation_functions/: Shows any representation function with eventual multiplicity at least g is realized by a set nearly as dense as a given bounded-multiplicity sequence.

cilleruelo_2014_infinite_sidon_sequences/: Gives the first explicit infinite Sidon sequence with counting function x to the power root two minus one, matching Ruzsa's nonconstructive record.

cilleruelo_2015_sidon_sets_asymptotic_bases/: Makes three advances on Erdos's conjecture that an infinite Sidon sequence can be an asymptotic basis of order three.

cilleruelo_2017_greedy_algorithm_b_h_g_sequences/: A modified greedy algorithm gives an infinite B_h[g] sequence whose nth term is at most 2g times n to the power h plus (h-1)/g.

cochrane_2026_mixed_incomplete_character_sums_rational_functions/: Strengthens the Graham-Ringrose bound for short character sums to nearly maximal smoothness and extends it to mixed sums of rational functions.

crocker_1971_sum_prime_two_powers_two/: Proves there are infinitely many odd integers that are not the sum of a prime and two powers of two.

croot_2026_combinatorial_large_sieve_sidon_sets_distances/: Develops a combinatorial large sieve giving the first super-polylogarithmic saving for Sidon sets in squares and new bounds for two grid-distance problems.

csajbok_2024_complete_3_term_arithmetic_progression_free/: Studies the minimum size of maximal 3-AP-free sets in finite vector spaces and odd cyclic groups, with square-root-order lower bounds matched up to constant factors in many vector spaces and for a dense set of moduli.

czerwinski_2023_sidon_sets_sum_free_sets_linear/: Improves the general upper bound on the largest Sidon set in the binary vector space F_2^t and classifies maximal Sidon sets in small dimensions.

daryxx_2026_erdos_problem_10_grechuk_partial_result/: Proves, with a write-up and a Lean 4 file the bounty site Conjectures.io kernel-verified and then set aside as a duplicate, that infinitely many even integers are not a prime plus at most three powers of 2, the Grechuk variant of Problem 10 and not the question itself.

ding_2020_green_s_problem_additive_complements_squares/: Confirms a conjecture of Chen and Fang by showing every additive complement of the squares falls below Green's critical profile by a linear amount infinitely often.

ding_2022_green_s_additive_complement_problem_k/: Extends the linear deviation obstruction for additive complements from the squares to k-th powers, with an explicit Gamma-function constant.

ding_2022_note_additive_complements_squares/: Proves the representation excess of any additive complement of the squares is at least 0.193 times the square root of N, improving Chen and Fang.

ding_2025_cross_representations_additive_complements_r_th/: Shows any additive complement of the r-th powers has representation excess at least of order N^(1-1/r), and N^(3/4-o(1)) for squares.

ding_2026_improved_upper_bound_ruzsa_number/: Proves the Ruzsa number satisfies R_m at most 128 for every modulus m, improving the previous bound of 192.

doorn_2025_smallest_set_such_that_every_positive/: Constructs a set such that every positive integer is a square plus one of its elements, with counting function below 2 phi^(5/2) sqrt(x), about 6.66 sqrt(x).

doorn_2026_completeness_exponentially_increasing_sequences/: Extends Graham's characterization of when the sequence of floors of t times alpha to the n is complete, settling all alpha at least the golden ratio.

erdos_1936_arithmetical_density_sum_two_sequences_one/: Proves the Schnirelmann density of a sequence plus a basis of order l is at least delta + delta(1-delta)/(2l).

erdos_1941_problem_sidon_additive_number_theory_related/: Brackets the largest Sidon set in {1,...,n} between (1/sqrt(2)-e)n^(1/2) and n^(1/2)+O(n^(1/4)), and shows that the number of representations as a sum of two terms cannot be eventually constant.

erdos_1954_results_additive_number_theory/: Builds a set with at most a constant times (log n)^2 elements up to n whose sums with the primes cover all large integers, and shows that (log n)^2 cannot be lowered for some sequences of positive lower density.

erdos_1956_problem_additive_number_theory/: Proves that the counting function of sums of two elements of a sequence cannot approximate cn with error o(n^{1/4} log^{-1/2-ε} n), the 1954 report's form of the Erdos-Fuchs theorem (the journal's has log^{-1/2}).

erdos_1960_additive_properties_random_sequences_positive_integers/: Shows random integer sequences of square-root-type density have sumsets of positive density with Poisson-distributed representation counts.

erdos_1961_representation_large_integers_as_sums_distinct/: Proves a sequence dense enough and hitting every arithmetic progression with distinct-term sums represents every sufficiently large integer.

erdos_1968_applications_graph_theory_number_theoretic_problems/: States that the largest set of integers up to n with all pairwise products distinct has pi(n) plus between two constant multiples of n^{3/4}/(log n)^{3/2} members, and proves the upper bound.

erdos_1977_bases_sets_integers/: Bounds the smallest basis B with A contained in B+B, showing most sets need a near-maximal basis while the squares need far fewer.

erdos_1979_systems_distinct_representatives_minimal_bases_additive/: An asymptotic basis of order two whose representation counts exceed c log n for some c > 1/log(4/3) contains a minimal asymptotic basis, and, if it also contains arbitrarily long intervals, a maximal asymptotic nonbasis.

erdos_1980_bases_exact_order/: A basis has an exact order precisely when the differences of consecutive elements have greatest common divisor one, and the exact order is of quadratic size.

erdos_1981_problems_results_additive_multiplicative_number_theory/: Poses problems on ratios of consecutive divisors, Sidon-type sets with almost all sums distinct, and gaps between squarefree numbers.

erdos_1982_problems_additive_number_theory/: Shows a system of sequences with distinct differences in [1,N] whose difference set exceeds (1+ε)N/2 must use many sequences, and poses a two-Sidon-set variant.

erdos_1984_extremal_problems_number_theory/: An ICM survey of Erdos's extremal problems on arithmetic progressions, Sidon sets and point distances, with many stated conjectures.

erdos_1988_partitions_bases_into_disjoint_unions_bases/: Shows an asymptotic basis of order h with at least c log n pairwise disjoint representations of each large n, c > 1/log(t^h/(t^h-1)), splits into t disjoint asymptotic bases of the same order.

erdos_1989_additive_bases_many_representations/: Gives an overlap condition on solution sets forcing an asymptotic basis of order two to contain a minimal one, and shows many representations do not suffice.

erdos_1990_representations_integers_as_sum_k_terms/: Proves that for every fixed k there is an asymptotic basis of order k in which the number of representations of n as a sum of k distinct elements is of order log n.

erdos_1994_sum_sets_sidon_sets_i/: Shows the sum set of a Sidon set is highly fragmented, with many blocks and large gaps, and poses nine unsolved problems on Sidon sets.

erdos_et_al_1995_sum_sets_sidon_sets_ii/: Proves a Sidon set in [1, N] has fewer than L/2 + 7L^(1/2)N^(1/4) sums in any length-L interval, builds an infinite Sidon set whose sum set has more than n^(1/3)/50 consecutive integers just after some m <= n for all large n, and bounds progression coverings of B_2[g] sets.

erdos_freud_1991_sums_sidon_sequence/: Erdős and Freud's 1991 paper on how many sums of a Sidon sequence in [1, n] fall below n: the theorem 1 - 1/sqrt 2 <= S(n)/n <= 1/pi asymptotically, the bounds 3/8 - eps <= T(n)/n <= 1/2 + eps for large n over arbitrary sets of at most (1 + o(1)) sqrt n elements (Proposition 1), the definition of quasi-Sidon sequences with a construction of size (2/sqrt 3) sqrt n and the unproved bound 1.98 sqrt n, and Proposition 2 with the remark that is the second question of Problem 14.

fabian_2019_strong_infinite_sidon_b_h_sets/: Improves lower bounds for the growth of strong infinite Sidon and B_h sets, proves an upper bound for strong B_h sets, and bounds B_h sets inside random infinite sets of integers from below.

fan_2026_strongly_complete_sets_conjecture_erdos/: Gives a strong-completeness criterion for sets with at least five elements in every dyadic interval and divergent sums of distances to integers, resolving Erdős's 1961 conjecture (Problem 254), and remarks that the sharp threshold two would imply Hegyvári's conjecture on the floors of the doubling multiples of two reals (Problem 354); context only for Problem 354.

fang_sandor_2022_function_sx_additive_complements/: Bounds the Erdős–Freud quantity SX(A,B) for additive complements: an exponent bound near the critical product ratio, the unconditional bound SX >= sqrt(1+C_0), and an exact formula for perfect complements.

fang_sandor_2022_sets_sum_difference_structure/: Classifies the pairs of sets of nonnegative integers that represent every nonnegative integer exactly once as a sum as the alternating digit sets of a mixed-radix expansion, and proves that such pairs have unique differences too.

folkman_1966_representation_integers_as_sums_distinct_terms/: Proves that a nondecreasing sequence with a_n <= M n^a, or a strictly increasing one with a_n <= M n^{1+a}, where 0 <= a < 1, is subcomplete, and complete when its subset sums meet every residue class.

gabdullin_2024_trigonometric_polynomials_frequencies_set_cubes/: Proves an L4-L2 bound for trigonometric polynomials with frequencies among cubes in a short interval, and that cubes in a shorter range form a Sidon set.

garaev_2026_sidon_sets_squares_cubes_quartics_short/: Finds, for the squares and for the cubes of the integers from N on, an endpoint in N below which they form a Sidon set for every N and which cannot be included for infinitely many N, and bounds the corresponding length for fourth powers between orders N^(3/5) and N^(12/13).

geneson_2026_deletion_thresholds_exponential_examples_complete_sequences/: Classifies the deletion thresholds of Problem 348 (only m at most 1), refutes Graham's conjecture that the floors of t times gamma to the n are complete for every t and every base below the golden ratio (Problem 349), and gives one such base with two coefficients of irrational ratio whose interleaved floor sequences are all even, hence incomplete: a negative answer to the every-base reading of the second question of Problem 354.

graham_1964_complete_sequences_polynomial_values/: Determines, by elementary means, the real polynomials f for which every sufficiently large integer is a sum of distinct values f(1), f(2), ...: the coefficients in the binomial basis must be rational, with positive leading coefficient and numerators of greatest common divisor 1.

graham_1964_property_fibonacci_numbers/: Proves that the sequence F_n - (-1)^n stays complete after deleting any finite subsequence and is not complete after deleting any infinite one, and that removing two terms from the Fibonacci sequence destroys completeness.

graham_1971_sums_integers_taken_fixed_sequence/: Graham's 1971 survey of sums of distinct terms from a fixed sequence of positive integers, from Sprague's complete sequences through Roth and Szekeres, Cassels, Erdős and Folkman to the thresholds of completeness, closing with twelve open questions: the origin of Problem 475 (distinct partial sums of residues modulo a prime), of Problem 354 (the floors of 2^n α and 2^n β) and of the real variant of Problem 1.

graham_nd_conjecture_erdos_additive_number_theory/: Determines for which pairs with 0 < t < 1 and 1 < a < 2 the sequence of integer parts of t times a to the n is complete, a region of area about 0.85.

granville_1998_binary_additive_problem_erdos_order_2_mod_p_2/: Ties Erdős's conjecture that every odd integer is a squarefree number plus a power of 2 to the primes p with p squared dividing 2 to the p minus 1 minus 1, in both directions, and proves conditional almost-all results with bounded numbers of powers of 2.

green_2001_number_squares_b_h_g_sets/: Proves a Fourier-analytic lower bound for additive energy of real-valued functions of fixed sum and deduces improved upper bounds for B_3, B_4 and B_2[g] sets.

green_2026_100_open_problems/: A personal collection of one hundred open problems in additive combinatorics and number theory, with commentary and updates on those since solved.

grekos_et_al_2003_erdos_turan_conjecture/: Reformulates the Erdős–Turán conjecture as the divergence of the finite quantity rho(x), the least possible maximal representation count of a basis of [0,x], and computes its first values.

habsieger_1995_additive_completion_polynomial_sets/: Improves the lower bound for additive complements of polynomial value sets, giving the constant (1-1/k)^{-1} sin(pi/k)/(pi/k), and 4/pi for squares.

habsieger_2016_numerical_note_upper_bounds_b_2/: Improves the best known upper bounds on the largest B_2[g] set in an interval of integers for g between 2 and 5.

hegyvari_1991_complete_sequences/: Proves the set of beta for which the Erdos-Graham sequence A_{alpha,beta} is incomplete is measurable and has Lebesgue measure either 0 or infinity.

hegyvari_1994_sumset_certain_sets/: Shows continuum many pairs make A_{alpha,beta} not even subcomplete, that for finite-dyadic alpha the normalized count g_alpha(m) of finite-dyadic partners making it complete tends to 1, and describes gaps in P(A_alpha).

hegyvari_2007_answer_question_burr_erdos_restricted_addition/: Answers a question of Burr and Erdős: sums of distinct elements of an asymptotic basis of order h have bounded gaps when h = 2, but need not when h >= 3.

hercher_2024_sum_squarefree_integers_power_two/: Verifies computationally that every odd integer below 2^50 is a squarefree number plus a power of two, using at most 2^13.

huber_2026_saturated_sidon_sets_consecutive_intervals_eight_mark_threshold_144/: Proves 144 is the largest interval length with a maximal Sidon set of exactly eight elements, by an explicit ruler and a computer-assisted exclusion, giving E156 an exact eight-element data point.

jain_2024_explicit_economical_additive_basis/: Gives an explicit additive basis of order two whose representation counts grow slower than any power, answering a question of Erdos.

jenw1n_2026_erdos_354_part_i_lean_proof/: Lean proof, accepted and paid by the bounty site Conjectures.io in September 2026, that for two positive reals with irrational ratio every sufficiently large integer is a sum of floors of their doubling multiples with each index used at most once, the first question of Problem 354 in the site's exact reading; the site's solution file is retained and was read as text, not built here, and no refereed publication exists.

jin_2014_density_versions_plunnecke_inequality/: Gives Jin's simplified proof of Plünnecke's Schnirelmann density bound and epsilon-delta variants for lower asymptotic and Banach densities.

kiss_2022_generalized_sidon_sets_perfect_powers/: Constructs sets of perfect k-th powers of nearly maximal density whose h-fold representation function stays bounded, for h = 2, for all large h, and for 2 <= h <= k under a Hypothesis-K-type bound.

kitamura_2026_lean_proof_erdos_problem_354_ii/: A Lean proof, published on GitHub on 5 September 2026 and not reviewed, of the formal-conjectures statement of the second question of Problem 354, which asks for some base between 1 and 2; the witness is the square root of the golden ratio, at which one floor sequence alone is already complete for every positive coefficient, so the pair and the irrational ratio play no role; a variant of the site's question, disputed as a misformalization.

kohayakawa_2015_number_sidon_sets_sparse_random_set_integers/: Bounds the number of Sidon subsets of an interval exponentially in its largest Sidon set and determines the size of the largest Sidon subset of a sparse random set of integers up to a factor n^{o(1)}.

kolountzakis_1996_additive_complements_primes_sets_similar_growth/: Builds an almost additive complement of the primes of size about log x log log x and shows prime-like sets can need complements of size log squared x.

kolountzakis_1996_density_b_h_g_sequences_minimum/: Bounds the size of B_h sets for even h via dense cosine sums and constructs unusually dense finite and infinite B_2[2] sequences.

kolountzakis_1999_uniform_distribution_residue_classes_dense_sets/: Shows dense Sidon subsets of an interval distribute nearly uniformly among residue classes, with explicit bounds on the discrepancy.

konieczny_2016_sets_recurrence_as_bases_positive_integers/: Answers Erdős's question negatively for sets of n with alpha n^2 near an integer being bases of order two, for sqrt2 and almost all alpha, and proves they are almost bases.

konstantoulas_2013_lower_bounds_conjecture_erdos_turan/: Proves that if the upper density of the integers missing from A+A is below 1/10 then r_A(n) > 5 for infinitely many n, a finite lower bound toward the Erdős–Turán conjecture.

konyagin_2009_erdos_turan_problem_infinite_groups/: Settles the Erdos-Turan representation problem for infinite abelian groups G with |2G| = |G| by determining exactly which possess a perfect additive basis of order two.

larsen_2026_robust_additive_bases_without_minimal_subbases/: Builds an additive basis whose representation counts grow logarithmically yet which contains no minimal subbasis, proving a conjecture of Erdos and Nathanson.

larsen_2026_three_questions_erdos_nathanson_asymptotic_bases/: Shows three robustness properties of asymptotic bases of order two are independent below the Erdos-Nathanson growth threshold.

lev_2004_reconstructing_integer_sets_representation_functions/: Gives one proof of Dombi's and Chen and Wang's partitions of the positive integers into two sets with equal sum-representation functions, partitions the positive integers into infinitely many perfect difference sets, and poses open problems.

lichtman_2024_modification_linear_sieve_count_twin_primes/: Modifies the linear sieve so its weights are strongly factorable, equidistributing primes to level x^(10/17) and improving the twin prime upper bound.

lindstrom_2000_b_h_g_sequences_b_h/: Shows that any B_h sequence yields a B_h[m^(h-1)] sequence m times as large, giving B_h[g] sets of size (gn)^(1/h)(1+o(1)) in an interval.

lorentz_1954_problem_additive_number_theory/: Proves that every infinite set of natural numbers has a density-zero additive complement, with an explicit bound in terms of its counting function.

ma_2022_note_additive_complements/: Studies additive complements whose product A(x)B(x) exceeds x by exactly 1 infinitely often, and exhibits a family of cluster points of the set of limsup values of A(x)B(x)/x that such pairs can attain.

ma_2026_largest_sidon_subsets_weak_sidon_sets/: Determines the largest guaranteed Sidon subset of a weak Sidon set exactly and improves both bounds for the analogous constant on (4,5)-sets.

martin_2005_constructions_generalized_sidon_sets/: Gives explicit constructions of large generalized Sidon sets, sets whose ordered pairwise sums repeat at most g times, with lower bounds for their square-root density and upper bounds for the analogue modulo n.

maynard_2020_primes_arithmetic_progressions_large_moduli_i/: Proves mean value theorems for primes in a fixed residue class to moduli beyond the square root of x, reaching moduli as large as x to the 11/21.

maynard_2020_primes_arithmetic_progressions_large_moduli_ii/: Extends mean value theorems with well-factorable weights for primes in a fixed residue class to triply well-factorable weights of level up to x to the 3/5, improving Bombieri-Friedlander-Iwaniec.

maynard_2020_primes_arithmetic_progressions_large_moduli_iii/: Proves mean value theorems for primes to moduli just past the square root of x that are fully uniform in the residue classes considered.

moser_1963_notes_number_theory/: Shows the average number of representations of an integer as a sum of consecutive primes tends to log 2, and lists related open questions.

nagy_2022_thin_sidon_sets_nonlinearity_vectorial_boolean/: Improves the vectorial nonlinearity lower bound via Sidon sets and builds complete Sidon sets of size q+2 from conics in characteristic two.

nagy_pach_tomon_2021_additive_bases_coset_covers/: Proves a strong weak additive-basis theorem over prime fields, gives an e-to-the-order-k-log-log-k bound for abelian coset covers, and extends non-vanishing linear-map results to several matrices.

narkiewicz_1960_remarks_conjecture_hanani_additive_number_theory/: Proves that if almost all integers have at least k representations as a sum from two sequences with limsup A(x)B(x)/x <= k, then almost all have exactly k and one of the sequences satisfies A(2x)/A(x) -> 1.

nash_1989_sequences/: Proves that the counting function of a B_4-sequence satisfies liminf A(n)(log n)^{1/4}/n^{1/4} finite, the B_4 analog of Erdos's B_2 bound.

nathanson_2003_generalized_additive_bases_konigs_lemma_erdos_turan_conjecture/: Defines generalized additive bases with prescribed orders and representation counts, characterizes when finite ones exist, and uses König's lemma to pass from finite bases to infinite ones.

nathanson_2013_additive_systems_theorem_de_bruijn/: Proves de Bruijn's theorem that every additive system decomposing the nonnegative integers is a British (mixed-radix) number system or a contraction of one.

nathanson_2014_paul_erdos_additive_bases/: Survey of Erdos's work on additive bases, covering Shnirelman density, the Erdos-Turan conjecture, and thin, minimal and maximal bases.

obryant_2004_complete_annotated_bibliography_work_related_sidon/: Surveys Sidon sequences and their generalizations with a consistent notation and supplies a complete annotated, hyperlinked bibliography of the literature.

obryant_2022_size_finite_sidon_sets/: Improves the upper bound for the largest Sidon set in {1,...,n} to fewer than n^(1/2) + 0.99703 n^(1/4) elements for large n.

obryant_2026_thickness_infinite_generalized_sidon_sets_i/: Proves a liminf upper bound for infinite g-Golomb rulers with constant 2·sqrt(g)/sqrt(log 2), and constructs one with large limsup counting function.

obryant_2026_thickness_infinite_generalized_sidon_sets_ii/: Gives an explicit constant in the liminf upper bound for the counting function of an infinite B_h set when h is even.

pan_2011_integers_not_form/: Proves unconditionally that the odd integers up to x not of the form p+2^a+2^b number at least x^(1-epsilon) for every epsilon.

pascadi_2025_exponents_distribution_primes_smooth_numbers/: Proves that primes with triply-well-factorable weights and smooth numbers are equidistributed in progressions to moduli up to x^(5/8-epsilon).

pikhurko_2006_dense_edge_magic_graphs_thin_additive/: Bounds edge-magic graph sizes and proves a quasi-Sidon subset of the first n integers has at most about 1.863 root n elements.

pilatte_2023_solution_erdos_sarkozy_sos_problem_asymptotic/: Proves that there is a Sidon set of natural numbers which is also an asymptotic basis of order three, answering a question of Erdos, Sarkozy and Sos.

plagne_2004_propos_de_la_fonction_d_erdos/: Sharpens both bounds on the Erdos-Graham function X(h) to floor(h(h+4)/3) <= X(h) <= h(h+1)/2 + ceil((h-1)/3).

plagne_nd_recent_progress_finite_b_h_g/: Survey of upper and lower bounds for the largest B_h[g] set in an interval of N integers, with new lower bounds for small g.

pliego_2024_erdos_turan_conjecture_growth_b_2/: Constructs, for each g at least 2, a B_2[g] sequence with counting function at least a constant times x^(g/(2g+1)) that is a basis of order three with one summand O(n^(1/g) (log n)^(2+1/g)).

price_2026_counterexample_erdos_problem_346/: Builds an integer sequence with the two Erdős-Graham deletion properties and ratios at least 6/5 whose consecutive ratios do not converge, which the paper presents as a negative answer to Problem 346.

rafik_2026_computational_evidence_erdos_problem_158_via/: Computes the first 2000 terms of the greedy B_2[2] sequence and tabulates its normalized counting function as numerical evidence on the liminf question.

rechnitzer_2026_first_128_digits_autoconvolution_inequality/: Computes rigorous bounds pinning down the first 128 digits of the L2 autoconvolution constant that enters upper bounds for B_2[g] sets.

redman_2021_small_maximal_sidon_set_z_2/: Constructs a maximal Sidon set in the group Z_2^n of size O((n times 2^n)^(1/3)), the group analog of Ruzsa's small maximal Sidon set in the integers.

riblet_2026_existence_sidon_set_distinct_distance_constant/: Proves a compactness property of Sidon sets, giving one whose reciprocal sum attains the distinct distance constant, and sharpens that constant's bounds.

romanoff_1934_uber_einige_satze_der_additiven/: Proves that the integers expressible as a prime plus a k-th power (k fixed), and as a prime plus a power of a fixed base, have positive lower density.

ruzsa_1985_note_additive_bases_integers/: Builds, for every order h at least 3, a basis of density zero whose (h-1)-fold sumset has counting function within a constant factor of the basis's own for arbitrarily large x, and proves that A_3(3x)/A(x) tends to infinity for every density-zero basis.

ruzsa_1990_just_basis/: Constructs an additive basis of order two whose representation counts are bounded in square mean, answering a problem related to the Erdős-Turán question on bases with bounded representation counts.

ruzsa_1998_additive_completion_primes/: Constructs sets with counting function O(log x) whose sums with the primes have lower density above 1 - eps, and proves a lower bound of order log x when the sums miss very few integers.

ruzsa_1998_small_maximal_sidon_set/: Constructs a maximal Sidon set in the first N integers of size at most a constant times the cube root of N log N.

ruzsajr_1972_problem_p/: Constructs a sequence of counting function at most a constant times x over log x such that every large integer is a power of 2 plus a term of the sequence.

sarkozy_1997_additive_representation_functions/: Surveys the regularity and value distribution of additive representation functions, proves a few new results, and collects related open problems.

saxton_2015_hypergraph_containers/: Builds a small family of containers covering all independent sets of a hypergraph and derives counting, coloring and extremal consequences.

shparlinski_2012_modular_hyperbolas/: Surveys results on the distribution and geometry of points on the modular hyperbola xy congruent to a mod m, with applications and open problems.

silva_2005_maximal_sidon_sets_matroids/: Shows that inside a finite generalized Sidon set of order (2h-1, h-1), h >= 2, the B_h-subsets form a matroid, so all maximal B_h-subsets have equal size.

singer_1938_theorem_finite_projective_geometry_some_applications_number_theory/: Constructs for every prime power m a perfect difference set of m+1 residues modulo m^2+m+1 by cycling a finite projective plane, a cyclic Sidon set of size about N^(1/2), larger than E156 seeks.

svyable_2026_infinite_deletions_strongly_minimal_additive_bases/: Claims a complete answer to a deletion question on strongly minimal additive bases: yes for order one, no for every order at least two.

tafula_2026_infinite_sidon_type_sets_zero_sum/: Proves density theorems limiting infinite sets with few representations by zero-sum linear forms, recovering Chen's theorem for even-order Sidon sequences.

thornburgh_2024_uniform_exclude_distributions_sidon_sets/: Shows graphs of APN plateaued functions with unbalanced components have uniform exclude distributions, and computes those of Gold and Kasami functions for even n.

turturean_2026_negative_answer_erdos_problem_870/: Constructs, for every order k at least 3 and every constant C, an additive basis with at least C log n representations of each large integer n and no minimal subbasis.

white_2024_optimal_l2_autoconvolution_inequality/: Pins the minimal squared L2 norm of an autoconvolution to within four millionths and improves upper bounds for several B_h[g] sets.

yu_2008_note_b_2_g_sets/: Improves the upper bound for the largest B_2[g] set in an interval to roughly the square root of 1.74217(2g-1)N for every g at least two.

yu_chen_2026_erdos_problem_354_i_strong_completeness_two_dyadic_floor_sequences/: An unrefereed manuscript of 13 September 2026 with its own Lean formalization claiming that for two positive reals with irrational ratio the nonzero floors of their doubling multiples form a strongly complete set, which implies the first question of Problem 354; later than the bounty site's accepted proof, listed as a proof claim on erdosproblems.com, and not reviewed here.

zhai_1999_additive_completion_kth_powers/: Shows a complement to the kth powers confined to a short initial interval must have at least about k times N to the power one minus one over k elements.

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