I want to know if 2 binary strings $s$ and $t$ each of $d$ length (dimension) and N = 2 (the alphebet) in this case 0 and 1 are similar to each other or not using the following distance function where $k$ denotes the iteration number or the time index
Using programming language, I implemented the formula but I am unable to decide if there should be a threshold so that distances less than the threshold are considered similar to each other. The QUestion stems from the tutorial https://www.math.ubc.ca/~andrewr/620341/pdfs/symb_sum.pdf which mentions another distance function. For both of the distance metric, what is the usual practice of deciding the bound $\epsilon$ in the condition $d(s,t) < \epsilon$
I am having a tough time in understanding the document and not sure if the Authors mean that 2 strings are similar if their distance is less than $1/2^n$ where $n$ is the number of iterations / length of the vector. Shall be greateful for an easier explanation.
How to find the minimum and maximum distance bound? Thank you
It looks like that document considers strings of infinite length. So the theorem says that if two strings agree on the first $n$ characters, their distance is less than $1/2^n$. If you're dealing with finite strings of length $n$, the smallest distance possible between two distinct strings is $1/2^n$, so in this case if the distance between two strings is less than $1/2^n$, the two strings are the same.