Waring-Hilbert problem on Cantor sets

On arithmetic properties of Cantor sets

“…In Section 6, we prove that for each integer m ≥ 3, there is a positive integer k ≤ 2 m+8 , such that {z : z ∈ C, |z| ≤ 1} ⊆ {z m 1 + • • • + z m k : z j ∈ W, 1 ≤ j ≤ k}, where W = {x + iy : x, y ∈ C} ∼ = C × C (see Theorem 6.2). This is another conjecture of Guo (see [2,Section 6]).…”

“…Corollary 2.2.1 and Corollary 2.3.1 of [2] are used frequently in our Section 3. For completeness, we state and prove them as follows.…”

“…So we have cos 5π 4 − mβ ∈ ( 1 5 , 1). Let x = 2 3 , y = r n 0 +1 and ρ = 4 9 + r 2n 0 +2 ∈ ( 2 3 , 1). We can see that…”

“…Recently, Guo studied analogs of Waring's problem on Cantor sets in [2], which was called the Waring-Hilbert problem. Let…”

“…In 2019, Athreya, Reznick and Tyson studied arithmetic of Cantor set (see [1]) and conjectured that every element in [0, 1] can be written as x 2 1 + x 2 2 + x 2 3 + x 2 4 with each x j in C. This conjecture was proved by Wang, Jiang, Li and Zhao in [13]. In fact, they considered general middle- 1 α Cantor sets C α obtained by removing (successively) the middle one open interval with length 1 α of the original for α > 1 and showed that…”