On a Conjecture of Cusick Concerning the Sum of Digits of $n$ and $n+t$

Abstract

Abstract. For a nonnegative integer t, let ct be the asymptotic density of natural numbers n for which s(n + t) ≥ s(n), where s(n) denotes the sum of digits of n in base 2. We prove that ct > 1/2 for t in a set of asymptotic density 1, thus giving a partial solution to a conjecture of T. W. Cusick stating that ct > 1/2 for all t. Interestingly, this problem has several equivalent formulations, for example that the polynomial X(X + 1) · · · (X + t − 1) has less than 2 t zeros modulo 2 t+1 . The proof of the main result is based on Chebyshev's inequality and the asymptotic analysis of a trivariate rational function using methods from analytic combinatorics.

Divisibility of binomial coefficients by powers of two

“…Note that the denominator H is identical to the denominator in [9]. Thus we can follow the known computations.…”

“…In order to prove the needed closeness property, we consider the random variable n → Θ(λ − s 2 (n) + u, n) − Φ(u/ √ λ)n. By bounding the second moment, using a procedure similar to the method used by Drmota, Kauers and Spiegelhofer [9] (see also [31]), we obtain an upper bound of the difference for all but few n.…”

“…We use the same idea as before performing the summation (here in two variables) later. But as we are dealing with a function in three variables, it is more suitable to use the techniques of [9,Proposition 4.3]. We only sketch the main differences here.…”

“…(We are aware of the theory of Pemantle and Wilson [25], which provides a method to obtain asymptotics of certain multivariate functions. However we noted in [9] that the trivariate function considered there turns out to be a limit case which has to be treated separately. As this function possesses, up to a factor, the same denominator as our function M (2) , the same restriction applies here.)…”

“…In this paper we prove that the family j → Θ(j, n) usually follows a normal distribution. The method used for proving this theorem involves the computation of first and second moments of Θ(j, n), and uses asymptotic analysis of multivariate generating functions by complex analytic methods, building on earlier work by Drmota (1994) and Drmota, Kauers and Spiegelhofer (2016).…”