论文标题
广义Collatz PX+1地图的平价 - 矢量参数化的傅立叶分析
Fourier Analysis of the Parity-Vector Parameterization of the Generalized Collatz px+1 maps
论文作者
论文摘要
令P为奇数素数，并考虑将整数X发送到X/2或（PX+1）/2的MAP H_P，具体取决于X是偶数还是奇数。地图x/2和（px+1）/2的任意组成序列的x = 0的值可以在2个ADIC整数（Z2）上进行参数化，从而导致从Z2到ZP的连续函数，作者称之为H_P的H_P的“特征函数”（或“ numen”）。当P-1是2和2的功率时，给出了Lipschitz-type的特征函数，这表明H_P的周期点等于在Z2上由特征函数实现的（合理的）整数值集。另外，尽管在特征函数下R的前图像在Z2中的HAAR度量为零，并且通过使用适当选择的Z2的自我安装来预发特征函数，但可以对上述复合材料进行傅立叶分析。使用这种方法，计算出明确的上限，以均衡向量的周期点的绝对值至少包含CEIL（ln（p）/ln2）-1零在任何两个连续之间
Let p be an odd prime, and consider the map H_p which sends an integer x to either x/2 or (px+1)/2 depending on whether x is even or odd. The values at x=0 of arbitrary composition sequences of the maps x/2 and (px+1)/2 can be parameterized over the 2-adic integers (Z2) leading to a continuous function from Z2 to Zp which the author calls the "characteristic function" (or "numen") of H_p. Lipschitz-type estimates are given for the characteristic function when p-1 is a power of 2 and 2 is a primitive root mod p, and it is shown that the set of periodic points of H_p is equal to the set of (rational) integer values attained by the characteristic function over Z2. Additionally, although the pre-image of R under the characteristic function has zero Haar measure in the Z2, by pre-composing the characteristic function with an appropriately selected self-embedding of Z2, one can perform Fourier analysis of the aforementioned composite. Using this approach, explicit upper bounds are computed for the absolute value of a periodic point of H_p whose parity vector contains at least ceil(ln(p)/ln2)-1 zeroes between any two consecutive ones
