论文标题
N+T的二进制数字
The binary digits of n+t
论文作者
论文摘要
二进制数字函数$ s $计数非负整数二进制扩展中的数量。对于任何非负整数$ t $，T。〜w。〜CUSICK定义了整数的渐近密度$ c_t $ $ n \ geq 0 $，以至于2011年，他在2011年\ [s（n+t）\ geq s（n）。他在2011年$ c_t> 1/2 $ nof n n n n n n n n n n n n n n n n n n n n not a $ sm forsect fornesty的差数很高。在本文中，我们证明存在明确的常量$ m_0 $，因此，如果$ t $的二进制扩展包含至少$ c_t> 1/2 $，则至少包含$ m_0 $最大的连续性块，仅留下“初始案例” - 几乎没有打开“初始案例” - 几乎没有最大值的块。此外，我们提高了Emme和Hubert（2019）的结果，证明差异$ s（n+t）-s（n）$根据高斯分布的行为，最大趋势趋于$ 0 $，作为$ t $ grows的二进制扩展中的最大块数量。
The binary sum-of-digits function $s$ counts the number of ones in the binary expansion of a nonnegative integer. For any nonnegative integer $t$, T.~W.~Cusick defined the asymptotic density $c_t$ of integers $n\geq 0$ such that \[s(n+t)\geq s(n).\] In 2011, he conjectured that $c_t>1/2$ for all $t$ -- the binary sum of digits should, more often than not, weakly increase when a constant is added. In this paper, we prove that there exists an explicit constant $M_0$ such that indeed $c_t>1/2$ if the binary expansion of $t$ contains at least $M_0$ maximal blocks of contiguous ones, leaving open only the "initial cases" -- few maximal blocks of ones -- of this conjecture. Moreover, we sharpen a result by Emme and Hubert (2019), proving that the difference $s(n+t)-s(n)$ behaves according to a Gaussian distribution, up to an error tending to $0$ as the number of maximal blocks of ones in the binary expansion of $t$ grows.
