On irregularities of distribution of weighted sums-of-digits

Let s ( n ) be the sum-of-digits function of n in base 2 . Newman (1969) [9] has shown the surprising phenomenon that # { 0 ≤ n < N ∣ s ( 3 n ) ≡ 0 mod 2 } − N 2 > c ⋅ N log 3 log 4 for all N (with some c > 0 ). ted by investigations of distribution properties of Niederreiter–Halton sequences we study # { 0 ≤ n < N ∣ s γ ( 3 n + k ) ≡ 0 mod 2 } − N 2 for k = 0 , 1 , 2 and γ = ( γ 0 , γ 1 , γ 2 , … ) ∈ { 0 , 1 } N 0 , where s γ ( n ) is the γ -weighted sum-of-digits function of n in base 2. We completely classify all γ and k for which Newman-type results hold, thereby generalising the result of Newman and results given by Drmota and Stoll (2008) [2]. We point out consequences of our results for the distribution of Niederreiter–Halton sequences (Corollary 1).