ON ASYMPTOTIC BASES WHICH HAVE DISTINCT SUBSET SUMS
نویسندگان
چکیده
Abstract Let k and l be positive integers satisfying $k \ge 2, 1$ . A set $\mathcal {A}$ of is an asymptotic basis order if every large enough integer can represented as the sum terms from About 35 years ago, P. Erdős asked: does there exist where all subset sums with at most are pairwise distinct exception a finite number cases long $l \le - ? We use probabilistic tools to prove existence $2k+1$ for which elements except ‘small’ numbers.
منابع مشابه
Integer Sets with Distinct Subset - Sums
In Section 1 we introduce the problem of finding minimal-height sets of n natural numbers with distinct subset-sums (SSD), and in Section 2 review the well-known Conway-Guy sequence u, conjectured to yield a minimal SSD set for every n. We go on (Section 3) to prove that u certainly cannot be improved upon by any "greedy" sequence, to verify numerically (Section 4) that it does yield SSD sets f...
متن کاملA Construction for Sets of Integers with Distinct Subset Sums
A set S of positive integers has distinct subset sums if there are 2|S| distinct elements of the set {∑ x∈X x : X ⊂ S } . Let f(n) = min{maxS : |S| = n and S has distinct subset sums}. Erdős conjectured f(n) ≥ c2n for some constant c. We give a construction that yields f(n) < 0.22002 · 2n for n sufficiently large. This now stands as the best known upper bound on f(n).
متن کاملSubset Sums
IAl > ((l/k) + E)n then there is a subset B L A such that 0 < 1 BI 0, let snd(nt) denote the smallest integer that does not divide PI. We prove that for every I-: > 0 there is a constant c = ~(8:) z I, such that for every n > 0 and every rn, n ' +' 6 WI < n'llog'n...
متن کاملOn distinct sums and distinct distances
The paper (Discrete Comput. Geom. 25 (2001) 629) of Solymosi and Tóth implicitly raised the following arithmetic problem. Consider n pairwise disjoint s element sets and form all ð 2Þn sums of pairs of elements of the same set. What is the minimum number of distinct sums one can get this way? This paper proves that the number of distinct sums is at least ns ; where ds 1⁄4 1=cJs=2n is defined in...
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: Bulletin of The Australian Mathematical Society
سال: 2021
ISSN: ['0004-9727', '1755-1633']
DOI: https://doi.org/10.1017/s0004972721000174