Monochromatic Schur Triples in Randomly Perturbed Dense Sets of Integers
نویسندگان
چکیده
منابع مشابه
On the Monochromatic Schur Triples Type Problem
We discuss a problem posed by Ronald Graham about the minimum number, over all 2-colorings of [1, n], of monochromatic {x, y, x + ay} triples for a ≥ 1. We give a new proof of the original case of a = 1. We show that the minimum number of such triples is at most n 2 2a(a2+2a+3) + O(n) when a ≥ 2. We also find a new upper bound for the minimum number, over all r-colorings of [1, n], of monochrom...
متن کاملOn the Minimum Number of Monochromatic Generalized Schur Triples
The solution to the problem of finding the minimum number of monochromatic triples (x, y, x + ay) with a > 2 being a fixed positive integer over any 2-coloring of [1, n] was conjectured by Butler, Costello, and Graham (2010) and Thanathipanonda (2009). We solve this problem using a method based on Datskovsky’s proof (2003) on the minimum number of monochromatic Schur triples (x, y, x + y). We d...
متن کاملEscape from attracting sets in randomly perturbed systems.
The dynamics of escape from an attractive state due to random perturbations is of central interest to many areas in science. Previous studies of escape in chaotic systems have rather focused on the case of unbounded noise, usually assumed to have Gaussian distribution. In this paper, we address the problem of escape induced by bounded noise. We show that the dynamics of escape from an attractor...
متن کاملOn Arithmetic Structures in Dense Sets of Integers
We prove that if A ⊆ {1, . . . , N } has density at least (log log N )−c, where c is an absolute constant, then A contains a triple (a, a+d, a+2d) with d = x2+ y2 for some integers x, y, not both zero. We combine methods of T. Gowers and A. Sárközy with an application of Selberg’s sieve. The result may be regarded as a step toward establishing a fully quantitative version of the polynomial Szem...
متن کاملBalanced Subset Sums in Dense Sets of Integers
Let 1 ≤ a1 < a2 < · · · < an ≤ 2n − 2 denote integers. Assuming that n is large enough, we prove that there exist ε1, . . . , εn ∈ {−1,+1} such that |ε1 + · · ·+εn| ≤ 1 and |ε1a1+ · · ·+εnan| ≤ 1. This result is sharp, and in turn it confirms a conjecture of Lev. We also prove that when n is even, every integer in a large interval centered at (a1 + a2 + · · · + an)/2 can be represented as the s...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: SIAM Journal on Discrete Mathematics
سال: 2019
ISSN: 0895-4801,1095-7146
DOI: 10.1137/18m1227007