A note on the Tu-Deng conjecture

Abstract

Let k be a positive integer. For any positive integer x = Σ ^∞ _i=0 x _i 2ⁱ, where x _i = 0, 1, we define the weight w(x) of x by w(x) ≔ Σ ^∞ _i=0 x _i . For any integer t with 0 < t < 2^k − 1, let S _t ≔ {(a, b) ∈ ℤ²|a + b ≡ t (mod 2^k − 1), w(a) + w(b) < k, 0 ≤ a, b ≤ 2^k − 2}. This paper gives explicit formulas for cardinality of S _t in the cases of w(t) ≤ 3 and an upper bound for cardinality of S _t when w(t) = 4. From this one then concludes that a conjecture proposed by Tu and Deng in 2011 is true if w(t) ≤ 4.