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Zeitschriftenartikel:

M. Drmota, M. Kauers, L. Spiegelhofer:
"On a Conjecture of Cusick Concerning the Sum of Digits of \$n\$ and \$n+t\$";
SIAM Journal on Discrete Mathematics, 30 (2016), 2; S. 621 - 649.

Kurzfassung englisch:
For a nonnegative integer \$t\$, let \$c_t\$ be the asymptotic density of natural numbers \$n\$ for which \$s(n+t)\geq s(n)\$, where \$s(n)\$ denotes the sum of digits of \$n\$ in base \$2\$. We prove that \$c_t>1/2\$ for \$t\$ in a set of asymptotic density \$1\$, thus giving a partial solution to a conjecture of Cusick stating that \$c_t > 1/2\$ for all \$t\$. Interestingly, this problem has several equivalent formulations, for example that the polynomial \$X(X+1)\cdots (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.