On the nonvanishing of representation functions of some special sequences

Abstract

a b s t r a c t For a given positive integer N, and any coloring function c : N → {0, 1} satisfying c(2k) = 1 − c(k), c(2k + 1) = c(k) for all k ≥ N, we show that for all n ≥ 20N, n has both a monochromatic representation and a multicolored representation, in other words, there exist x, y, u, v ∈ N, such that n = x+y = u+v, c(x) = c(y) and c(u) ̸ = c(v). Similar results are obtained for another kind of coloring function c : N → {0, 1} satisfying c(2k) = c(k) and c(2k + 1) = 1 − c(k) for all k ≥ N. This answers a question of Y.-G. Chen on the values of representation functions.

Sets With Almost Coinciding Representation Functions

“…The investigation of the partitions of the set of nonnegative integers with identical representation functions was a popular topic in the last few decades [1], [3], [4], [5], [7], [9], [11], [12], [13]. It is easy to see that R Theorem 1 (Nathanson, 1978).…”

Partitions of the set of nonnegative integers with the same representation functions

“…The set A is called Thue-Morse sequence. The investigation of the partitions of the set of nonnegative integers with identical representation functions was a popular topic in the last few decades [1], [2], [7], [8], [9]. By using the Thue -Morse sequence in 2002 Dombi [5] constructed two sets of nonnegative integers with infinite symmetric difference such that the corresponding representation functions are identical.…”