1. Run a spell-checker.
2. If you use $$X$$ or $$i$$ as variables, don’t write them as X or i.
3. Write $$\log(n)$$ instead of $$log(n)$$, $$\arg\max$$ instead of $$argmax$$, $$\Pr(E)$$ instead of $$Pr(E)$$), etc. You can use the “\operatorname” command for non-standard operators.
4. For inner product notation, use $$\langle x,y \rangle$$ instead of $$<x,y>$$.
5. Make sure all parentheses, brackets, curly braces, etc. are matched, and also properly sized, e.g., $\left( \frac{x+y}{2} \right)$ rather than $( \frac{x+y}{2} .$
6. Be consistent in your use of calligraphic (e.g., $$\mathcal{X}$$), blackboard (e.g., $$\mathbb{P}$$), and other font styles.
7. If you have a “tall” mathematical expression like $$\left( \sum_{x=0}^{2^n-1} \frac{2^{2^x}}{3} \right)$$, it is better to put it in a separate equation display, like $\left( \sum_{x=0}^{2^n-1} \frac{2^{2^x}}{3} \right) .$
8. If you need to use some text expression inside an equation, use the “\text” command, e.g., $\{ x \in [0,1] : \text{1/x is a prime integer} \} .$
9. Mathematical expressions and equations should generally be written as if part of a complete sentence. For example, I define a set $$S$$ by $S := \{ x \in [0,1] : \text{1/x is a prime integer} \} ,$ and there is a comma at the end of the display to separate the independent clauses.
10. In $$\LaTeX$$, it is better to use “double-backtick” for opening quotation marks and “double-apostrophe” for closing quotation marks.
11. Make sure notation and jargon is properly defined before use.
12. Define “theorem” environments as needed (e.g., using the amsthm package).