Proceedings of Integers Conference 2009 LARGE ZERO - FREE SUBSETS OF Z / p Z Jean - Marc Deshouillers Institut
نویسندگان
چکیده
A finite subset A of an abelian group G is said to be zero-free if the identity element of G cannot be written as a sum of distinct elements from A. In this article we study the structure of zero-free subsets of Z/pZ, the cardinalities of which are close to the largest possible. In particular, we determine the cardinality of the largest zero-free subset of Z/pZ, when p is a sufficiently large prime. —To Mel Nathanson and Carl Pomerance, with friendship and respect.
منابع مشابه
Ja n 20 09 Large zero - free subsets of Z / p Z
A finite subset A of an abelian group G is said to be zero-free if the identity element of G cannot be written as a sum of distinct elements from A. In this article we study the structure of zero-free subsets of Z/pZ the cardinality of which is close to largest possible. In particular, we determine the cardinality of the largest zero-free subset of Z/pZ, when p is a sufficiently large prime. Fo...
متن کاملOn sum-free sets modulo p
Let p be a sufficiently large prime and A be a sum-free subset of Z/pZ; improving on a previous result of V. F. Lev, we show that if |A| = card(A) > 0.324p, then A is contained in a dilation of the interval [|A| , p− |A|] (mod. p).
متن کاملSelection of a Large Sum-Free Subset in Polynomial Time
A set E of integers is called sum-free if x + y 6 = z for all x; y; z 2 E. Given a set A = fn 1 ; : : :; n N g of integers we show how to extract a sum-free subset E of A with jEj > N=3. The algorithm requires polynomial time in the size of the input. A subset E of an additive group is called sum-free if x+y 6 = z for all x; y; z 2 E. Erd} os 3] and Alon and Kleitman 1] have proved that every s...
متن کاملThe number of maximal sum-free subsets of integers
Cameron and Erdős [6] raised the question of how many maximal sum-free sets there are in {1, . . . , n}, giving a lower bound of 2bn/4c. In this paper we prove that there are in fact at most 2(1/4+o(1))n maximal sum-free sets in {1, . . . , n}. Our proof makes use of container and removal lemmas of Green [8, 9] as well as a result of Deshouillers, Freiman, Sós and Temkin [7] on the structure of...
متن کاملOn Sums of Primes and Triangular
We study whether sufficiently large integers can be written in the form cp+ Tx, where p is either zero or a prime congruent to r mod d, and Tx = x(x + 1)/2 is a triangular number. We also investigate whether there are infinitely many positive integers not of the form (2ap−r)/m+Tx with p a prime and x an integer. Besides two theorems, the paper also contains several conjectures together with rel...
متن کامل