scholarly journals The number of distinct subset sums of a finite set of vectors

1993 ◽  
Vol 63 (2) ◽  
pp. 234-256
Author(s):  
Ernest Brickell ◽  
Michael Saks
Keyword(s):  
Distinct Subset ◽  
Finite Set ◽  
Subset Sums

scholarly journals Integer sets with distinct subset-sums

1988 ◽  
Vol 50 (181) ◽  
pp. 297-297 ◽  
Author(s):  
W. F. Lunnon
Keyword(s):  
Distinct Subset ◽  
Subset Sums

scholarly journals A construction for sets of integers with distinct subset sums

10.37236/1341 ◽  
1997 ◽  
Vol 5 (1) ◽  
Author(s):  
Tom Bohman
Keyword(s):  
Upper Bound ◽  
Distinct Subset ◽  
Subset Sums ◽  
Positive Integers

A set S of positive integers has distinct subset sums if there are $2^{|S|}$ distinct elements of the set $\left\{ \sum_{x \in X} x: X \subset S \right\} . $ Let $$f(n) = \min\{ \max S: |S|=n {\rm \hskip2mm and \hskip2mm} S {\rm \hskip2mm has \hskip2mm distinct \hskip2mm subset \hskip2mm sums}\}.$$ Erdős conjectured $ f(n) \ge c2^{n}$ for some constant c. We give a construction that yields $f(n) < 0.22002 \cdot 2^{n}$ for n sufficiently large. This now stands as the best known upper bound on $ f(n).$

A Note on the Erdös Distinct Subset Sums Problem

2021 ◽  
Vol 35 (1) ◽  
pp. 322-324
Author(s):  
Quentin Dubroff ◽  
Jacob Fox ◽  
Max Wenqiang Xu
Keyword(s):  
Distinct Subset ◽  
Subset Sums

scholarly journals ON ASYMPTOTIC BASES WHICH HAVE DISTINCT SUBSET SUMS

2021 ◽  
pp. 1-7
Author(s):  
SÁNDOR Z. KISS ◽  
VINH HUNG NGUYEN
Keyword(s):  
Positive Integer ◽  
Finite Number ◽  
Distinct Subset ◽  
Subset Sums ◽  
Asymptotic Basis ◽  
Positive Integers

Abstract Let k and l be positive integers satisfying $k \ge 2, l \ge 1$ . A set $\mathcal {A}$ of positive integers is an asymptotic basis of order k if every large enough positive integer can be represented as the sum of k terms from $\mathcal {A}$ . About 35 years ago, P. Erdős asked: does there exist an asymptotic basis of order k where all the subset sums with at most l terms are pairwise distinct with the exception of a finite number of cases as long as $l \le k - 1$ ? We use probabilistic tools to prove the existence of an asymptotic basis of order $2k+1$ for which all the sums of at most k elements are pairwise distinct except for ‘small’ numbers.

Non-repetitive sequences

1970 ◽  
Vol 68 (2) ◽  
pp. 267-274 ◽  
Author(s):  
P. A. B. Pleasants
Keyword(s):  
Repetitive Sequences ◽  
Finite Set ◽  
Infinite Sequences

This note is concerned with infinite sequences whose terms are chosen from a finite set of symbols. A segment of such a sequence is a set of one or more consecutive terms, and a repetition is a pair of finite segments that are adjacent and identical. A non-repetitive sequence is one that contains no repetitions.

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