广义Collatz px+1映射的奇偶向量参数化的傅里叶分析
Fourier Analysis of the Parity-Vector Parameterization of the Generalized Collatz px+1 maps
AI总结：
本文针对广义Collatz px+1映射H_p，提出基于2进整数的特征函数参数化方法，推导其Lipschitz型估计与周期点性质，并通过Z2自嵌入结合傅里叶分析，得到特定奇偶向量对应的周期点绝对值的显式上界。
摘要：
设p为奇素数，考虑映射H_p：当整数x为偶数时将其映射为x/2，为奇数时映射为(px+1)/2。由x/2和(px+1)/2构成的任意复合序列在x=0处的取值可通过2进整数（Z2）进行参数化，由此得到一个从Z2到Zp的连续函数，作者将其称为H_p的“特征函数”（或“numen”）。当p-1是2的幂且2是模p的原根时，本文给出了该特征函数的Lipschitz型估计，并证明H_p的周期点集等于特征函数在Z2上取得的（有理）整数值集合。此外，尽管特征函数下R的原像在Z2中的Haar测度为零，但通过将特征函数与一个适当选取的Z2自嵌入进行前置复合，可以对上述复合函数进行傅里叶分析。利用该方法，本文计算了H_p的周期点绝对值的显式上界，这类周期点的奇偶向量在任意两个连续的1之间至少包含⌈ln(p)/ln2⌉-1个0。
英文摘要：
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
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