On completely multiplicative ± 1 $\pm 1$ sequences that omit many consecutive + 1 $+1$ values

IF 0.9 3区数学 Q2 MATHEMATICS

Bulletin of the London Mathematical Society Pub Date : 2025-03-31 DOI:10.1112/blms.70056

Yichen You

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Abstract

We say that $\pm 1$ -valued completely multiplicative functions are length- $k$ functions $f$ if they take the value $+ 1$ at at most $k$ consecutive integers. We introduce a method to extend the length of $f$ using the idea of the “rotation trick” in [7]. Under the assumption of Elliott's conjecture, this method allows us to construct length- $k$ functions systematically for $k ⩾ 4$ which generalizes the work of Schur for $k = 2$ and Hudson for $k = 3$ .

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在省略许多连续的+1$ +1$值的完全乘法±1$ \pm 1$序列上

我们说±1 $\pm 1$值的完全乘法函数是长度为- k的$k$函数f $f$如果它们取+ 1的值$+1$最多k个$k$连续整数。我们引入了一种方法，利用[7]中的“旋转技巧”来扩展f $f$的长度。在艾略特猜想的假设下，该方法允许我们系统地为k或4 $k\geqslant 4$构建长度- k $k$函数，它概括了k = 2的Schur的工作$k = 2$和Hudson的k = 3 $k =3$。

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