Insert constraints into estimates and error terms

Quick description

In analysis and analytic number theory, one sometimes obtains estimates for expressions involving several variables or parameters subject to various constraints concerning their respective sizes. If these estimates have to be carried along and transformed in substantial ways, or if many of them involving different constraints must be combined at some point, it may be useful (if it is possible) to incorporate the constraints inside the estimates by adding terms which make the estimate trivial if the constraints are not satisfied, and reduce to the useful estimate otherwise.

This is similar in spirit to the use of a characteristic function of a set to represent a summation or integration condition, as described in Getting rid of nasty cutoffs for example.

Prerequisites

Basic calculus or even simple combinatorics and sums.

General discussion

The idea of the trick is best explained with concrete elementary examples.

Example 1

Consider the inequality

$\sqrt{x}\leq x.$

This is only valid if $x\geq 1$ . However, if we write

$\sqrt{x}\leq x+1$

the inequality is valid for $x\geq 0$ , and is not asymptotically worse than the previous one when $x$ gets large.

Example 2

Suppose you have a function $S(n,p)$ depending on an integer $n\geq 1$ and a prime number $p$ , and that you know the upper bound

$|S(n,p)|\leq p$

for all $n$ , and

$|S(n,p)|\leq \sqrt{p}$

if $p$ does not divide $n$ . If these estimates are used extensively later on, with different values of $n$ and $p$ , sorting out which of the two applies may become a bookkeeping nightmare. Writing

$|S(n,p)|\leq \sqrt{(n,p)}\sqrt{p}$

where $(n,p)$ is the GCD of $n$ and $p$ encapsulates both statements neatly and is much easier to carry around and combine with other estimates.

This article is incomplete. More examples of this are planned (in particular more convincing ones).

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