I just remembered an idiomatic expression using "to within" that is somewhat common. Sometimes people will say something like this: "I came TO WITHIN a hair's breadth of _(doing just about anything)__________." like: "I came TO WITHIN a hair's breadth of hitting a home run in baseball!"

But you are right to ask about this Dolokhov, because it is a strange construction in English. "To within" just means, basically; Something-----comes----THAT-----close. Something comes to within THAT value range. I do not think I have ever seen the phrase used in any other context but measurement. In general English, which is to say outside of actual Mathematics, you might see that phrase in an expression like this: "I came to within an inch of getting killed by that car!" "I came to within an inch of persuading that girl to go to the movies with me." "I came to within an inch of getting rich in the stock market." .

(Correction on last sentence) "...the statistic APPROXIMATES a scale factor the square of the third moment."

Hi Dolokhov: "To within" as a phrase, means this. The statistic (or any mathematical value) shows us something. The mathematical value is a statistic which shows a tendencies. There is therefore, a scale of value. Normally, such scales can be reflected by Place Value. It is common to see decimal values of Tenths, Hundredths, and Thousandths. So, where the precision or accuracy is concerned, the phrase "to within" refers to the degree of accuracy. So sentences are written this way: The machine could cut the steel to within a Tenth of an Millimeter... (or) Hundredth of a Millimeter (or) a Ten Thousandth of a Millimeter. Your statistical data might be referred to this way in your sentence: "...the statistic is APPROXIMATES a scale factor the square of the third moment." (I hope that helps)

You're breaking that sentence down incorrectly. Think of it this way: to 'within a scale factor' the square of the third moment I'd like to see the rest of the context if you can. I can explain it better if I know what the numbers are talking about, but 'within a scale factor' likely means 'almost exactly the same'.