Combining two elementary proofs we decrease the gap between the upper and lower bounds by 0.2% in a classical combinatorial number theory problem. We show that the maximum size of a Sidon set of {1, 2, . . . , n} is at most √ n + 0.998n 1/4 for sufficiently large n.
HistoryIn 1932 S. Sidon asked a question of a fellow student P. Erdős. Their advisor was L. Fejér, an outstanding mathematician (cf. Fejér kernel) working on summability of infinite series, who had a number of outstanding students who contributed to mathematical analysis (M.
Using tools from o-minimality, we prove that for two bivariate polynomials P (x, y) and Q(x, y) with coefficients in R or C to simultaneously exhibit small expansion, they must exploit the underlying additive or multiplicative structure of the field in nearly identical fashion. This in particular generalizes the main result of Shen [24] and yields an Elekes-Ronyai type structural result for symmetric non-expanders, resolving an issue mentioned by de Zeeuw in [9]. Our result also places sum-product phenomena into a more general picture of model-theoretic interest.
The method of alternating projections involves projecting an element of a Hilbert space cyclically onto a collection of closed subspaces. It is known that the resulting sequence always converges in norm and that one can obtain estimates for the rate of convergence in terms of quantities describing the geometric relationship between the subspaces in question, namely their pairwise Friedrichs numbers. We consider the question of how best to order a given collection of subspaces so as to obtain the best estimate on the rate of convergence. We prove, by relating the ordering problem to a variant of the famous Travelling Salesman Problem, that correctness of a natural form of the Greedy Algorithm would imply that P = NP, before presenting a simple example which shows that, contrary to a claim made in the influential paper [9], the result of the Greedy Algorithm is not in general optimal. We go on to establish sharp estimates on the degree to which the result of the Greedy Algorithm can differ from the optimal result. Underlying all of these results is a construction which shows that for any matrix whose entries satisfy certain natural assumptions it is possible to construct a Hilbert space and a collection of closed subspaces such that the pairwise Friedrichs numbers between the subspaces are given precisely by the entries of that matrix.2010 Mathematics Subject Classification. 47J25, 65F10 (68Q25).
In this entry point into the subject, combining two elementary proofs, we decrease the gap between the upper and lower bounds by 0.2% in a classical combinatorial number theory problem. We show that the maximum size of a Sidon set of {1, 2, . . . , n} is at most √ n + 0.998n 1/4 for sufficiently large n.
HistoryIn 1932 S. Sidon asked a question of a fellow student P. Erdős. Their advisor was L. Fejér, an outstanding mathematician (cf. Fejér kernel) working on summability of infinite series, who had a number of outstanding students who contributed to mathematical analysis (M.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.