Question

The ring Z_n:={0,1,..,n-1} under addition and multiplication modulo n. Suppose a,b,c,x $\in$ Z_n are nonzero and the cyclic order R(a,b,c) holds, then under what conditions does R(ax,bx,cx) hold ?

Answer 1

Answer. Under the condition $$ \left\lbrace\frac{x(b-a)}n\right\rbrace<\left\lbrace\frac{x(c-a)}n\right\rbrace \qquad\qquad(*) $$ where $\lbrace\cdot\rbrace$ denotes the fractional part of a real number.

The condition $R(a,b,c)$ is equivalent to $R(0,b-a,c-a)$ and means that the least residues of $b-a$ and $c-a$ modulo $n$ are ordered in the increasing order as integers. Note that the least residue of an integer $m$ modulo $n$ can be expressed by means of $n\cdot\lbrace m/n\rbrace$.

The relations ($*$) are hardly related for different values of $x=1,2,\dots$: one can always construct $n$ and residues $a$ and $b$ in such a way that for a given range of $x\in X_1\cup X_2$ ($X_1$ and $X_2$ are two finite disjoint sets of $\mathbb Z_{>0}$) the inequality ($*$) hold for $x\in X_1$ and does not hold for $x\in X_2$.

Answer 2

There is some sort of theory for rational linear forms in fractional parts holding at all integer multiples prime to some denominator, by Dvornicich and Zannier: see Fractional parts of linear polynomials, Journal of the Australian Mathematical Society (2001), 70: 401-424. The place I know about where this issue comes up classically is the theory of the hypergeometric equation, namely Schwarz's list, as revisited by Landau and Errera. This application is mentioned in the paper, and is dealt with at length by a book of Matsuda.

In the nature of the whole discussion, one is trying to prove n is small if the fractional parts of several rational numbers with denominator n do not distribute themselves uniformly about under multiplication. It all looks like a messy combinatorial problem, but the example shows that sometimes there is a good or at least effective answer at the end.