Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.
Updated
Additive bases
L125_random_singer_lift_residual/: Uniform independent lifts of a perfect difference set leave linearly many integer targets unblocked with probability tending to one when p cubed is comparable to N. The proof uses bounded dependence and includes doubles and integer carries; it does not exclude structured lifts or small repairs.
L126_path_shell_resonance/: Every prescribed large path can be paired with admissible smaller paths giving logarithmically many distinct representations of one integer. One infinite path family realizes such peaks, so local increment restrictions alone cannot give a uniform bound against all earlier shells.
L208_two_parabola_maximal_sidon/: For every odd prime power q, a parabola in the plane z = 0 and a non-square twisted parabola in the plane x = 0 form a maximal strong Sidon set of 2q-1 elements in F_q^3, the cube-root scale of the group.
L209_singer_level_maximal_sidon/: A perfect difference set modulo q = p^2+p+1 placed at level zero, plus one greedily chosen point at each level of an abelian group H of odd order h <= p/4, is a maximal strong Sidon set of size p+h in Z_q x H; for cyclic H coprime to q this is a cyclic group of order about p^3/4.
L216_sidon_linear_capacity_barrier/: A genuine measure on fixed-size sets satisfies every ordinary difference and sum capacity in expectation up to sqrt(N)+N^(1/4)/2-10, obstructing single linear-capacity certificates below exponent 1/4.
L217_sidon_block_quartic_obstruction/: The rotated block law defeats linear Sidon capacities, but erasing forbidden moments and restoring each degree's total moment gives a negative quadratic-square expectation at degree four.
L239_consecutive_unique_differences/: If m integers represent every positive difference up to L exactly once, then L <= (3 + 2 sqrt(2)) m^(5/3) - 2m + 2, regardless of their diameter. Sliding interval counts and an elementary smoothing bound exclude a quadratic-length Sidon difference basis.
L272_translated_unique_differences/: If k integers have distinct nonzero ordered differences and their difference set contains s+(W-W), where |s| exceeds the diameter of W, then |W| <= (3/2 + sqrt(2)) k^(5/3). An averaged sliding count gives the bound for every finite nonempty W.
L273_sidon_parity_obstruction/: An alternating quadratic has negative square expectation after zeroing forbidden moments of block/uniform mixtures, with or without degreewise renormalization, once the excess above sqrt(N) exceeds an absolute constant in the stated even-block family.
L2_two_representation_exponent/: One infinite positive-integer set has at most two unordered representations of every sum and logarithmic lower counting exponent at least 3/7.
L33_dyadic_root_floor_completeness/: Two residue classes of exponents turn a single geometric floor sequence into the two rays of the accepted dyadic completeness theorem.
L3_square_complement_nine_halves/: An explicit positive-integer square complement has exact lower and upper counting limits 3 sqrt(2) and 9/2, with uniform real-cutoff estimates.
L4_random_modulus_exponent/: One infinite positive-integer set has at most two representations of each sum and counting exponent root six minus two with an explicit loss.
These claims state their representation conventions and asymptotic quantifiers explicitly. A lower growth exponent for one construction need not give positive density at that exponent, or settle a question about a larger exponent. Source results remain in the additive-bases library.