Nonnegative k-sums, fractional covers, and probability of small deviations

More than twenty years ago, Manickam, Mikl\'{o}s, and Singhi conjectured that for any integers $n, k$ satisfying $n \geq 4k$, every set of $n$ real numbers with nonnegative sum has at least $\binom{n-1}{k-1}$ $k$-element subsets whose sum is also nonnegative. In this paper we discuss the connection of this problem with matchings and fractional covers of hypergraphs, and with the question of estimating the probability that the sum of nonnegative independent random variables exceeds its expectation by a given amount. Using these connections together with some probabilistic techniques, we verify the conjecture for $n \geq 33k^2$. This substantially improves the best previously known exponential lower bound $n \geq e^{ck \log\log k}$. In addition we prove a tight stability result showing that for every $k$ and all sufficiently large $n$, every set of $n$ reals with a nonnegative sum that does not contain a member whose sum with any other $k-1$ members is nonnegative, contains at least $\binom{n-1}{k-1}+\binom{n-k-1}{k-1}-1$ subsets of cardinality $k$ with nonnegative sum.Comment: 15 pages, a section of Hilton-Milner type result adde

Applications of an exact counting formula in the Bousso-Polchinski Landscape

The Bousso-Polchinski (BP) Landscape is a proposal for solving the Cosmological Constant Problem. The solution requires counting the states in a very thin shell in flux space. We find an exact formula for this counting problem which has two simple asymptotic regime one of them being the method of counting low $\Lambda$ states given originally by Bousso and Polchinski. We finally give some applications of the extended formula: a robust property of the Landscape which can be identified with an effective occupation number, an estimator for the minimum cosmological constant and a possible influence on the KKLT stabilization mechanism.Comment: 43 pages, 11 figures, 2 appendices. We have added a new section (3.4) on the influence of the fraction of non-vanishing fluxes in the KKLT mechanism. Other minor changes also mad

Integral points of fixed degree and bounded height

By Northcott's Theorem there are only finitely many algebraic points in affine $n$-space of fixed degree over a given number field and of height at most $X$. For large $X$ the asymptotics of these cardinalities have been investigated by Schanuel, Schmidt, Gao, Masser and Vaaler, and the author. In this paper we study the case where the coordinates of the points are restricted to algebraic integers, and we derive the analogues of Schanuel's, Schmidt's, Gao's and the author's results.Comment: to appear in Int. Math. Res. Notice

Combinatorial and Additive Number Theory Problem Sessions: '09--'19

These notes are a summary of the problem session discussions at various CANT (Combinatorial and Additive Number Theory Conferences). Currently they include all years from 2009 through 2019 (inclusive); the goal is to supplement this file each year. These additions will include the problem session notes from that year, and occasionally discussions on progress on previous problems. If you are interested in pursuing any of these problems and want additional information as to progress, please email the author. See http://www.theoryofnumbers.com/ for the conference homepage.Comment: Version 3.4, 58 pages, 2 figures added 2019 problems on 5/31/2019, fixed a few issues from some presenters 6/29/201