UPPER AND LOWER BOUNDS FOR FINITE Bh[g] SEQUENCES
نویسندگان
چکیده
We give a non trivial upper bound, Fh(g, N), for the size of a Bh[g] subset of {1, ...., N} when g > 1. In particular, we prove F2(g, N) ≤ 1.864(gN) + 1 Fh(g, N) ≤ 1 (1 + cosh(π/h))1/h (hh!gN), h > 2. On the other hand we exhibit B2[g] subsets of {1, ...., N} with g + [g/2] √ g + 2[g/2] N + o(N) elements. 1. Upper bounds Let h ≥ 2, g ≥ 1 be integers. A subset A of integers is called a Bh[g]-sequence if for every positive integer m, the equation m = x1 + · · ·+ xh, x1 ≤ · · · ≤ xh, xi ∈ A has, at most, g distinct solutions. Let Fh(g,N) denote the maximum size of a Bh[g] sequence contained in [1, N ]. If A is a Bh[g] subset of {1, ..., N}, then (|A|+h−1 h ) ≤ ghN , which implies the trivial upper bound (1.1) Fh(g, N) ≤ (ghh!N) For g = 1, h = 2, it is possible to take advantage of counting the differences xi − xj instead of the sums xi + xj , because the differences are all distinct. In this way, P. Erdős and P. Turán [2] proved that F2(1, N) ≤ N + O(N), which is the best possible except for the estimate of error term. For h = 2m, Jia [4] proved F2m(1; N) ≤ (m(m!))N + O(N). A similar upper bound for F2m−1(1, N) has been proved independently by S.Chen [1] and S.W.Graham [3]: F2m−1(1, N) ≤ ((m!)2)1/2m−1N1/2m−1 + O(N1/4m−2). However, for g > 1, the situation is completely different because the same difference can appear many times, and, for g > 1 nothing better than (1.1) is known. In this paper we improve this trivial upper bound. Theorem 1.1. F2(g, N) ≤ 1.864(gN) + 1 Fh(g, N) ≤ 1 (1 + cosh(π/h))1/h (hh!gN) , h > 2
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