Additive Bases and Sidon Sets

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E0009/: Asks whether the odd integers that are not the sum of a prime and two powers of 2 have positive upper density.

E0010/: Asks whether some fixed k makes every large integer the sum of a prime and at most k powers of 2; open, with three powers shown insufficient for infinitely many even integers.

E0011/: Asks whether every large odd integer is the sum of a squarefree number and a power of 2.

E0014/: Asks whether the integers up to N lacking a unique representation as a sum of two elements of a set must number nearly the square root of N; yes, and never little-o of it, by two Lean proofs the bounty site Conjectures.io accepted.

E0016/: Asks whether the odd integers not of the form a power of 2 plus a prime form the union of an infinite arithmetic progression and a set of density zero.

E0028/: Asks whether a set whose sumset omits only finitely many integers must have integers with arbitrarily many representations as sums of two elements.

E0029/: Asks for an explicit set whose sumset is all natural numbers while the number of representations of n grows slower than every power of n.

E0030/: Asks whether the largest Sidon set in the first N integers has size the square root of N plus an error smaller than every power of N.

E0031/: Asks whether every infinite set of natural numbers has a density-zero companion whose sumset with it omits only finitely many integers.

E0032/: Asks how sparse a set can be if every large integer is a prime plus one of its members, measured against the square of the logarithm of N.

E0033/: Asks how sparse a set can be if every large integer is a square plus one of its members, measured against the square root of N.

E0035/: Asks whether adding an additive basis of order k to a set of Schnirelmann density alpha raises the density by at least alpha times one minus alpha over k.

E0039/: Asks whether an infinite Sidon set can contain nearly the square root of N elements up to N, for every positive tolerance.

E0040/: Asks which growth rates just below the square root of N force some integers to have arbitrarily many representations as sums of two elements.

E0041/: Asks whether an infinite set with all triple sums distinct must have its counting function up to N infinitely often much smaller than the cube root of N.

E0042/: Asks whether every Sidon set in the first N integers can be paired with a Sidon set of any fixed size whose difference set meets its own only at zero.

E0043/: Asks whether two Sidon sets in the first N integers whose difference sets meet only at zero together have at most as many pairs as a largest Sidon set plus a constant, and a constant fraction fewer when equal in size.

E0044/: Asks whether every Sidon set in the first N integers extends to a Sidon set in a longer interval that is nearly as large as the largest possible.

E0066/: Asks whether some set of naturals has its number of representations as a sum of two elements, divided by the logarithm of n, tending to a nonzero limit.

E0152/: Asks whether every sufficiently large finite Sidon set has arbitrarily many pairwise sums whose neighbors one above and one below are not pairwise sums.

E0153/: Asks whether the mean squared gap between consecutive elements of the sumset of a finite Sidon set grows without bound as the set grows.

E0154/: Asks whether the sumset of a near-maximal Sidon set inside the first N integers is evenly spread over small moduli, for instance half even and half odd.

E0155/: Asks whether the largest Sidon subset of the first N plus k integers exceeds that of the first N integers by at most one, for every fixed k and all large N.

E0156/: Asks whether the first N integers contain a maximal Sidon set of size only order N to the power one third.

E0157/: Asks whether there is an infinite Sidon set that is also an asymptotic basis of order three, so all large integers are sums of three of its elements.

E0158/: Asks whether every infinite integer set with at most two representations of each number as a sum of two elements has counting function whose ratio to root N has lower limit zero.

E0221/: Asks whether some set of integers with at most about N over log N elements up to N lets every large integer be a power of two plus one of its elements.

E0241/: Asks whether the largest subset of the first N integers whose three-element sums are all distinct has size asymptotic to the cube root of N.

E0326/: Asks whether a minimal basis of order two exists whose kth smallest element divided by k squared tends to a nonzero constant; Erdős first asked it for any basis of order two, which Cassels answered yes.

E0330/: Asks whether a minimal basis of positive density exists in which, for each of its elements, the integers needing that element have positive upper density.

E0333/: Asks whether every set of integers of density zero lies in the sumset of some set whose counting function is smaller than the square root of N.

E0336/: Determines the limit of h of r divided by r squared, where h of r is the largest finite exact order of an additive basis of order r.

E0337/: Asks whether every additive basis of density zero has its sumset counting function grow infinitely faster than its own counting function.

E0338/: Characterizes when a basis has a restricted order, using distinct summands only, and whether that order is bounded in terms of the ordinary order.

E0339/: Asks whether the integers that are sums of exactly r distinct elements of a basis of order r must have positive lower density.

E0340/: The growth rate of the greedy Sidon sequence, and whether its counting function is at least N to the power one half minus any epsilon.

E0343/: Asks, after Folkman, whether a multiset of integers with linear counting function must have subset sums containing an infinite arithmetic progression, in the form Szemerédi and Vu prove, with one absolute constant; Folkman's question for every constant is open.

E0344/: Asks whether every set of integers with at least a sufficiently large constant times the square root of N elements up to N, for every N, has subset sums containing an infinite arithmetic progression.

E0345/: Asks whether infinitely many k make the completeness threshold of the kth powers larger than that of the next powers.

E0346/: Asks whether a sequence complete after any finite deletion but never after an infinite one, with ratios bounded away from one, must have ratios tending to the golden ratio.

E0347/: Asks whether some integer sequence with successive ratios tending to two has subset sums of density one even after any finite set of terms is removed.

E0348/: Determines for which m less than n a complete sequence can stay complete after removing any m elements yet fail after removing any n elements.

E0349/: Determines for which positive t and alpha the integer parts of t times alpha to the n form a complete sequence, distinct terms summing to all large integers.

E0351/: Asks whether the numbers p of n plus one over n, for a rational polynomial p with positive leading coefficient, stay complete after any finite set is removed.

E0354/: Asks whether the rounded-down doubling multiples of two reals with irrational ratio form a complete sequence, and the same for a base between one and two; the first question is answered yes, the second has two readings.

E0358/: Asks whether some infinite sequence of integers writes every large n as a sum of consecutive terms at least twice, or in ways tending to infinity.

E0425/: Estimates the largest subset of the first n integers with distinct pairwise products, and bounds sets whose products of r increasing members are distinct.

E0530/: The order of the largest Sidon subset, one with no non-trivial equal pairwise sums, guaranteed inside every set of N real numbers.

E0707/: Asks whether every finite Sidon set of integers can be extended to a perfect difference set modulo p squared plus p plus 1 for some prime p.

E0757/: Determines the best constant c such that n reals whose every four-element subset has at least eleven differences contain a Sidon set of size c times n.

E0772/: The largest Sidon set guaranteed inside any n integers in which no number has more than k representations as a sum of two elements.

E0773/: The size of the largest Sidon subset of the squares up to N squared, and whether it is N to the power one minus o(1).

E0840/: Determines how fast the largest quasi-Sidon subset of the integers up to N grows, where a set is quasi-Sidon if its sumset is nearly as large as possible.

E0861/: Asks how many Sidon subsets of the integers up to N there are, compared with two to the power of the largest Sidon set size.

E0862/: Asks whether the number of maximal Sidon subsets of the integers up to N is below two to the power o(square root of N), and whether it exceeds two to the power N^c for some c > 0.

E0863/: Asks whether, for r at least 2, the largest sets up to N with at most r representations of each sum, and of each positive difference, have different square-root constants, and whether the difference constant is the smaller.

E0864/: Estimates the largest set of integers up to N with at most one number having more than one representation as a sum of two members.

E0868/: Asks whether an additive basis of order two whose representation counts tend to infinity must contain a minimal such basis.

E0869/: Asks whether the union of two disjoint additive bases of order two must contain a minimal additive basis of order two.

E0870/: Asks whether representation counts growing like a constant times the logarithm force an additive basis of order k to contain a minimal one, for k at least 3.

E0871/: Asks whether an additive basis of order two whose representation counts tend to infinity can be split into two disjoint additive bases of order two.

E0880/: Asks whether the integers representable as sums of k or fewer distinct members of an additive basis of order k have bounded gaps.

E0881/: Concerns additive bases of order k that are minimal, in the sense that removing any infinite subset destroys the basis property.

E1145/: Asks a question about two increasing sets of positive integers whose nth elements have ratio tending to one.

E1147/: Asks whether the set of n with alpha times n squared within one over the logarithm of n of an integer is an additive basis of order two, for irrational alpha.

E1191/: Asks whether every infinite Sidon set has counting function o((x/log x)^(1/2)) along a subsequence, and whether some infinite Sidon set keeps it at least x^(1/2)/(log x)^c for some c > 0.

E1192/: Concerns the possible growth of the number of representations of n as a sum of r elements of a set of natural numbers.

E1194/: For a set in which every positive integer n is uniquely a difference a_n - b_n of two members, asks how fast a_n/n must grow, where a_n is the larger member of the representation of n.

E1206/: Asks whether the set of cubes up to N cubed contains a Sidon set, with all pairwise sums distinct, of size proportional to N.

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Representation functions of sets of integers: additive bases and the Erdos-Turan basis conjecture, Sidon and B_h sets in which sums or differences are almost distinct, and complete sequences whose subset sums represent every large integer.

Site tags routed here: additive basis, additive combinatorics, complete sequences, distances, geometry, irrational, number theory, primes, sidon sets, squares.