$\require{mathtools}
%
%%% GENERIC MATH %%%
%
% Environments
\newcommand{\al}[1]{\begin{align}#1\end{align}} % need this for \tag{} to work
\renewcommand{\r}{\mathrm}
%
% Greek
\newcommand{\eps}{\epsilon}
\newcommand{\veps}{\varepsilon}
\let\fi\phi % because it looks like an f
\let\phi\varphi % because it looks like a p
%
% Miscellaneous shortcuts
% .. over and under
\newcommand{\ss}[1]{_{\substack{#1}}}
\newcommand{\ob}{\overbrace}
\newcommand{\ub}{\underbrace}
\newcommand{\ol}{\overline}
\newcommand{\tld}{\widetilde}
\newcommand{\HAT}{\widehat}
\newcommand{\f}{\frac}
\newcommand{\s}[2]{#1 /\mathopen{}#2}
\newcommand{\rt}{\sqrt}
% .. relations
\newcommand{\sr}{\stackrel}
\newcommand{\sse}{\subseteq}
\newcommand{\ce}{\coloneqq}
\newcommand{\ec}{\eqqcolon}
\newcommand{\ap}{\approx}
\newcommand{\ls}{\lesssim}
\newcommand{\gs}{\greatersim}
% .. miscer
\newcommand{\q}{\quad}
\newcommand{\qq}{\qquad}
\newcommand{\heart}{\heartsuit}
%
% Delimiters
% (I needed to create my own because the MathJax version of \DeclarePairedDelimiter doesn't have \mathopen{} and that messes up the spacing)
% .. one-part
\newcommand{\p}[1]{\mathopen{}\left( #1 \right)}
\newcommand{\b}[1]{\mathopen{}\left[ #1 \right]}
\newcommand{\set}[1]{\mathopen{}\left\{ #1 \right\}}
\newcommand{\abs}[1]{\mathopen{}\left\lvert #1 \right\rvert}
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\newcommand{\ceil}[1]{\mathopen{}\left\lceil #1 \right\rceil}
\newcommand{\inner}[1]{\mathopen{}\left\langle #1 \right\rangle}
% .... (use phantom to force at least the standard height of double bars)
\newcommand{\norm}[1]{\mathopen{}\left\lVert #1 \vphantom{f} \right\rVert}
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%% .. two-part
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\newcommand{\at}[2]{\left.#1\right|_{#2}}
% ..... (use phantom to force at least the standard height of double bar)
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%\newcommand{\para}[2]{#1\vphantom{f} \mathop{}\middle\|\mathop{} #2}
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\newcommand{\ppa}[2]{\p{\para{#1}{#2}}}
\newcommand{\bpa}[2]{\b{\para{#1}{#2}}}
%\newcommand{\bpaco}[4]{\bpa{\incond{#1}{#2}}{\incond{#3}{#4}}}
\newcommand{\bpaco}[4]{\bpa{\cond{#1}{#2}}{\cond{#3}{#4}}}
%
% Levels of closeness
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\newcommand{\sdot}[1]{\sr{.}{#1}}
\newcommand{\slog}[1]{\sr{\log}{#1}}
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\newcommand{#3}{\mathrel{#1}}
\newcommand{#4}{\mathrel{#1\!\!#1}}
\newcommand{#5}{\mathrel{#1\!\!#1\!\!#1}}
\newcommand{#6}{\mathrel{(\sdot{#1})}}
\newcommand{#7}{\mathrel{(\slog{#1})}}
}
\let\lt\undefined
\let\gt\undefined
% .. vanilla versions (is it within a constant?)
\newcommand{\ez}{\scirc=}
\newcommand{\eq}{\simeq}
\newcommand{\eqq}{\mathrel{\eq\!\!\eq}}
\newcommand{\eqqq}{\mathrel{\eq\!\!\eq\!\!\eq}}
\newcommand{\lez}{\scirc\le}
\newcommand{\lq}{\preceq}
\newcommand{\lqq}{\mathrel{\lq\!\!\lq}}
\newcommand{\lqqq}{\mathrel{\lq\!\!\lq\!\!\lq}}
\newcommand{\gez}{\scirc\ge}
\newcommand{\gq}{\succeq}
\newcommand{\gqq}{\mathrel{\gq\!\!\gq}}
\newcommand{\gqqq}{\mathrel{\gq\!\!\gq\!\!\gq}}
\newcommand{\lz}{\scirc<}
\newcommand{\lt}{\prec}
\newcommand{\ltt}{\mathrel{\lt\!\!\lt}}
\newcommand{\lttt}{\mathrel{\lt\!\!\lt\!\!\lt}}
\newcommand{\gz}{\scirc>}
\newcommand{\gt}{\succ}
\newcommand{\gtt}{\mathrel{\gt\!\!\gt}}
\newcommand{\gttt}{\mathrel{\gt\!\!\gt\!\!\gt}}
% .. dotted versions (is it equal in the limit?)
\newcommand{\ed}{\sdot=}
\newcommand{\eqd}{\sdot\eq}
\newcommand{\eqqd}{\sdot\eqq}
\newcommand{\eqqqd}{\sdot\eqqq}
\newcommand{\led}{\sdot\le}
\newcommand{\lqd}{\sdot\lq}
\newcommand{\lqqd}{\sdot\lqq}
\newcommand{\lqqqd}{\sdot\lqqq}
\newcommand{\ged}{\sdot\ge}
\newcommand{\gqd}{\sdot\gq}
\newcommand{\gqqd}{\sdot\gqq}
\newcommand{\gqqqd}{\sdot\gqqq}
\newcommand{\ld}{\sdot<}
\newcommand{\ltd}{\sdot\lt}
\newcommand{\lttd}{\sdot\ltt}
\newcommand{\ltttd}{\sdot\lttt}
\newcommand{\gd}{\sdot>}
\newcommand{\gtd}{\sdot\gt}
\newcommand{\gttd}{\sdot\gtt}
\newcommand{\gtttd}{\sdot\gttt}
% .. log versions (is it equal up to log?)
\newcommand{\elog}{\slog=}
\newcommand{\eqlog}{\slog\eq}
\newcommand{\eqqlog}{\slog\eqq}
\newcommand{\eqqqlog}{\slog\eqqq}
\newcommand{\lelog}{\slog\le}
\newcommand{\lqlog}{\slog\lq}
\newcommand{\lqqlog}{\slog\lqq}
\newcommand{\lqqqlog}{\slog\lqqq}
\newcommand{\gelog}{\slog\ge}
\newcommand{\gqlog}{\slog\gq}
\newcommand{\gqqlog}{\slog\gqq}
\newcommand{\gqqqlog}{\slog\gqqq}
\newcommand{\llog}{\slog<}
\newcommand{\ltlog}{\slog\lt}
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\newcommand{\glog}{\slog>}
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%
% Miscellaneous
\newcommand{\LHS}{\mathrm{LHS}}
\newcommand{\RHS}{\mathrm{RHS}}
% .. operators
\DeclareMathOperator{\poly}{poly}
\DeclareMathOperator{\polylog}{polylog}
\DeclareMathOperator{\quasipoly}{quasipoly}
\DeclareMathOperator{\negl}{negl}
\DeclareMathOperator*{\argmin}{arg\,min}
\DeclareMathOperator*{\argmax}{arg\,max}
% .. functions
\DeclareMathOperator{\id}{id}
\DeclareMathOperator{\sign}{sign}
\DeclareMathOperator{\err}{err}
\DeclareMathOperator{\ReLU}{ReLU}
% .. analysis
\let\d\undefined
\newcommand{\d}{\operatorname{d}\mathopen{}}
\newcommand{\df}[2]{\f{\d #1}{\d #2}}
\newcommand{\ds}[2]{\s{\d #1}{\d #2}}
\newcommand{\part}{\partial}
\newcommand{\partf}[2]{\f{\part #1}{\part #2}}
\newcommand{\parts}[2]{\s{\part #1}{\part #2}}
\newcommand{\grad}[1]{\mathop{\nabla\!_{#1}}}
% .. sets
\newcommand{\N}{\mathbb{N}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\F}{\mathbb{F}}
\newcommand{\zo}{\set{0,1}}
\newcommand{\pmo}{\set{\pm 1}}
%
%%% SPECIALIZED MATH %%%
%
% Logic
\renewcommand{\and}{\wedge}
\newcommand{\AND}{\bigwedge}
\newcommand{\or}{\vee}
\newcommand{\OR}{\bigvee}
\newcommand{\xor}{\oplus}
\newcommand{\XOR}{\bigoplus}
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\DeclareMathOperator{\1}{\mathbb{1}}
%
% Linear algebra
\renewcommand{\span}{\mathrm{span}}
\DeclareMathOperator{\rank}{rank}
\DeclareMathOperator{\proj}{proj}
\DeclareMathOperator{\dom}{dom}
\DeclareMathOperator{\Img}{Im}
\newcommand{\transp}{\mathsf{T}}
\renewcommand{\t}{^\transp}
% ... named tensors
\newcommand{\namedtensorstrut}{\vphantom{fg}} % milder than \mathstrut
\newcommand{\name}[1]{\mathsf{\namedtensorstrut #1}}
\newcommand{\nbin}[2]{\mathbin{\underset{\substack{#1}}{\namedtensorstrut #2}}}
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\newcommand{\ndef}[2]{\newcommand{#1}{\name{#2}}}
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%
% Probability
\newcommand{\Normal}{\mathcal{N}}
\let\Pr\undefined
\DeclareMathOperator*{\Pr}{Pr}
\DeclareMathOperator*{\G}{\mathbb{G}}
\DeclareMathOperator*{\Odds}{Od}
\DeclareMathOperator*{\E}{E}
\DeclareMathOperator*{\Var}{Var}
\DeclareMathOperator*{\Cov}{Cov}
\DeclareMathOperator*{\corr}{corr}
\DeclareMathOperator*{\median}{median}
\newcommand{\dTV}{d_{\mathrm{TV}}}
\newcommand{\dHel}{d_{\mathrm{Hel}}}
\newcommand{\dJS}{d_{\mathrm{JS}}}
% ... information theory
\let\H\undefined
\DeclareMathOperator*{\H}{H}
\DeclareMathOperator*{\I}{I}
\DeclareMathOperator*{\D}{D}
%
%%% SPECIALIZED COMPUTER SCIENCE %%%
%
% Complexity classes
% .. classical
\newcommand{\Poly}{\mathsf{P}}
\newcommand{\NP}{\mathsf{NP}}
\newcommand{\PH}{\mathsf{PH}}
\newcommand{\PSPACE}{\mathsf{PSPACE}}
\renewcommand{\L}{\mathsf{L}}
% .. probabilistic
\newcommand{\formost}{\mathsf{Я}}
\newcommand{\RP}{\mathsf{RP}}
\newcommand{\BPP}{\mathsf{BPP}}
\newcommand{\MA}{\mathsf{MA}}
\newcommand{\AM}{\mathsf{AM}}
\newcommand{\IP}{\mathsf{IP}}
\newcommand{\RL}{\mathsf{RL}}
% .. circuits
\newcommand{\NC}{\mathsf{NC}}
\newcommand{\AC}{\mathsf{AC}}
\newcommand{\ACC}{\mathsf{ACC}}
\newcommand{\TC}{\mathsf{TC}}
\newcommand{\Ppoly}{\mathsf{P}/\poly}
\newcommand{\Lpoly}{\mathsf{L}/\poly}
% .. resources
\newcommand{\TIME}{\mathsf{TIME}}
\newcommand{\SPACE}{\mathsf{SPACE}}
\newcommand{\TISP}{\mathsf{TISP}}
\newcommand{\SIZE}{\mathsf{SIZE}}
% .. keywords
\newcommand{\co}{\mathsf{co}}
\newcommand{\Prom}{\mathsf{Promise}}
%
% Boolean analysis
\newcommand{\harpoon}{\!\upharpoonright\!}
\newcommand{\rr}[2]{#1\harpoon_{#2}}
\newcommand{\Fou}[1]{\widehat{#1}}
\DeclareMathOperator{\Ind}{\mathrm{Ind}}
\DeclareMathOperator{\Inf}{\mathrm{Inf}}
\DeclareMathOperator{\Der}{\mathrm{D}}
\DeclareMathOperator{\Stab}{\mathrm{Stab}}
\DeclareMathOperator{\T}{T}
\DeclareMathOperator{\sens}{\mathrm{s}}
\DeclareMathOperator{\bsens}{\mathrm{bs}}
\DeclareMathOperator{\fbsens}{\mathrm{fbs}}
\DeclareMathOperator{\Cert}{\mathrm{C}}
\DeclareMathOperator{\DT}{\mathrm{DT}}
\DeclareMathOperator{\CDT}{\mathrm{CDT}} % canonical
\DeclareMathOperator{\ECDT}{\mathrm{ECDT}}
\DeclareMathOperator{\CDTv}{\mathrm{CDT_{vars}}}
\DeclareMathOperator{\ECDTv}{\mathrm{ECDT_{vars}}}
\DeclareMathOperator{\CDTt}{\mathrm{CDT_{terms}}}
\DeclareMathOperator{\ECDTt}{\mathrm{ECDT_{terms}}}
\DeclareMathOperator{\CDTw}{\mathrm{CDT_{weighted}}}
\DeclareMathOperator{\ECDTw}{\mathrm{ECDT_{weighted}}}
\DeclareMathOperator{\AvgDT}{\mathrm{AvgDT}}
\DeclareMathOperator{\PDT}{\mathrm{PDT}} % partial decision tree
\DeclareMathOperator{\DTsize}{\mathrm{DT_{size}}}
\DeclareMathOperator{\W}{\mathbf{W}}
% .. functions (small caps sadly doesn't work)
\DeclareMathOperator{\Par}{\mathrm{Par}}
\DeclareMathOperator{\Maj}{\mathrm{Maj}}
\DeclareMathOperator{\HW}{\mathrm{HW}}
\DeclareMathOperator{\Th}{\mathrm{Th}}
\DeclareMathOperator{\Tribes}{\mathrm{Tribes}}
\DeclareMathOperator{\RotTribes}{\mathrm{RotTribes}}
\DeclareMathOperator{\CycleRun}{\mathrm{CycleRun}}
\DeclareMathOperator{\SAT}{\mathrm{SAT}}
\DeclareMathOperator{\UniqueSAT}{\mathrm{UniqueSAT}}
%
% Dynamic optimality
\newcommand{\OPT}{\mathsf{OPT}}
\newcommand{\Alt}{\mathsf{Alt}}
\newcommand{\Funnel}{\mathsf{Funnel}}
%
% Alignment
\DeclareMathOperator{\Amp}{\mathrm{Amp}}
%
%%% TYPESETTING %%%
%
% In text
\renewcommand{\th}{^{\mathrm{th}}}
\newcommand{\degree}{^\circ}
%
% Fonts
% .. bold
\newcommand{\BA}{\boldsymbol{A}}
\newcommand{\BB}{\boldsymbol{B}}
\newcommand{\BC}{\boldsymbol{C}}
\newcommand{\BD}{\boldsymbol{D}}
\newcommand{\BE}{\boldsymbol{E}}
\newcommand{\BF}{\boldsymbol{F}}
\newcommand{\BG}{\boldsymbol{G}}
\newcommand{\BH}{\boldsymbol{H}}
\newcommand{\BI}{\boldsymbol{I}}
\newcommand{\BJ}{\boldsymbol{J}}
\newcommand{\BK}{\boldsymbol{K}}
\newcommand{\BL}{\boldsymbol{L}}
\newcommand{\BM}{\boldsymbol{M}}
\newcommand{\BN}{\boldsymbol{N}}
\newcommand{\BO}{\boldsymbol{O}}
\newcommand{\BP}{\boldsymbol{P}}
\newcommand{\BQ}{\boldsymbol{Q}}
\newcommand{\BR}{\boldsymbol{R}}
\newcommand{\BS}{\boldsymbol{S}}
\newcommand{\BT}{\boldsymbol{T}}
\newcommand{\BU}{\boldsymbol{U}}
\newcommand{\BV}{\boldsymbol{V}}
\newcommand{\BW}{\boldsymbol{W}}
\newcommand{\BX}{\boldsymbol{X}}
\newcommand{\BY}{\boldsymbol{Y}}
\newcommand{\BZ}{\boldsymbol{Z}}
\newcommand{\Ba}{\boldsymbol{a}}
\newcommand{\Bb}{\boldsymbol{b}}
\newcommand{\Bc}{\boldsymbol{c}}
\newcommand{\Bd}{\boldsymbol{d}}
\newcommand{\Be}{\boldsymbol{e}}
\newcommand{\Bf}{\boldsymbol{f}}
\newcommand{\Bg}{\boldsymbol{g}}
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\newcommand{\Bi}{\boldsymbol{i}}
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\newcommand{\Bk}{\boldsymbol{k}}
\newcommand{\Bp}{\boldsymbol{p}}
\newcommand{\Bq}{\boldsymbol{q}}
\newcommand{\Br}{\boldsymbol{r}}
\newcommand{\Bs}{\boldsymbol{s}}
\newcommand{\Bt}{\boldsymbol{t}}
\newcommand{\Bu}{\boldsymbol{u}}
\newcommand{\Bv}{\boldsymbol{v}}
\newcommand{\Bw}{\boldsymbol{w}}
\newcommand{\Bx}{\boldsymbol{x}}
\newcommand{\By}{\boldsymbol{y}}
\newcommand{\Bz}{\boldsymbol{z}}
\newcommand{\Balpha}{\boldsymbol{\alpha}}
\newcommand{\Bbeta}{\boldsymbol{\beta}}
\newcommand{\Bgamma}{\boldsymbol{\gamma}}
\newcommand{\Bdelta}{\boldsymbol{\delta}}
\newcommand{\Beps}{\boldsymbol{\eps}}
\newcommand{\Bveps}{\boldsymbol{\veps}}
\newcommand{\Bzeta}{\boldsymbol{\zeta}}
\newcommand{\Beta}{\boldsymbol{\eta}}
\newcommand{\Btheta}{\boldsymbol{\theta}}
\newcommand{\Biota}{\boldsymbol{\iota}}
\newcommand{\Bkappa}{\boldsymbol{\kappa}}
\newcommand{\Blambda}{\boldsymbol{\lambda}}
\newcommand{\Bmu}{\boldsymbol{\mu}}
\newcommand{\Bnu}{\boldsymbol{\nu}}
\newcommand{\Bxi}{\boldsymbol{\xi}}
\newcommand{\Bomicron}{\boldsymbol{\omicron}}
\newcommand{\Bpi}{\boldsymbol{\pi}}
\newcommand{\Brho}{\boldsymbol{\rho}}
\newcommand{\Bsigma}{\boldsymbol{\sigma}}
\newcommand{\Btau}{\boldsymbol{\tau}}
\newcommand{\Bupsilon}{\boldsymbol{\upsilon}}
\newcommand{\Bphi}{\boldsymbol{\phi}}
\newcommand{\Bfi}{\boldsymbol{\fi}}
\newcommand{\Bchi}{\boldsymbol{\chi}}
\newcommand{\Bpsi}{\boldsymbol{\psi}}
\newcommand{\Bomega}{\boldsymbol{\omega}}
% .. calligraphic
\newcommand{\CA}{\mathcal{A}}
\newcommand{\CB}{\mathcal{B}}
\newcommand{\CC}{\mathcal{C}}
\newcommand{\CD}{\mathcal{D}}
\newcommand{\CE}{\mathcal{E}}
\newcommand{\CF}{\mathcal{F}}
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\newcommand{\CJ}{\mathcal{J}}
\newcommand{\CK}{\mathcal{K}}
\newcommand{\CL}{\mathcal{L}}
\newcommand{\CM}{\mathcal{M}}
\newcommand{\CN}{\mathcal{N}}
\newcommand{\CO}{\mathcal{O}}
\newcommand{\CP}{\mathcal{P}}
\newcommand{\CQ}{\mathcal{Q}}
\newcommand{\CR}{\mathcal{R}}
\newcommand{\CS}{\mathcal{S}}
\newcommand{\CT}{\mathcal{T}}
\newcommand{\CU}{\mathcal{U}}
\newcommand{\CV}{\mathcal{V}}
\newcommand{\CW}{\mathcal{W}}
\newcommand{\CX}{\mathcal{X}}
\newcommand{\CY}{\mathcal{Y}}
\newcommand{\CZ}{\mathcal{Z}}
% .. typewriter
\newcommand{\TA}{\mathtt{A}}
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\newcommand{\TT}{\mathtt{T}}
\newcommand{\TU}{\mathtt{U}}
\newcommand{\TV}{\mathtt{V}}
\newcommand{\TW}{\mathtt{W}}
\newcommand{\TX}{\mathtt{X}}
\newcommand{\TY}{\mathtt{Y}}
\newcommand{\TZ}{\mathtt{Z}}$
Miscellaneous facts from Ryan O’Donnell’s “TCS toolkit” class (and some more).
Asymptotic tricks
Harmonic numbers. $H_n \coloneqq \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} \sim \ln n$. In fact, $\ln(n+1) \leq H_n = \ln(n) + \gamma - O(\frac{1}{n})$ where $\gamma \approx 0.577$ is the Euler–Mascheroni constant.
Math of small things. For $0 \le \epsilon \le \f12$,
- $e^\epsilon, \frac{1}{1-\epsilon} = 1 + \epsilon + O(\epsilon^2)$ and $\leq 1+2\epsilon$;
- $e^{-\epsilon}, \frac{1}{1+\epsilon} = 1 - \epsilon + O(\epsilon^2)$ and $\leq 1-\frac{\epsilon}{2}$;
- $\ln(1+\epsilon) = \epsilon - O(\epsilon^2)$ and $\geq \frac{\epsilon}{2}$;
- $\ln\p{\frac{1}{1-\epsilon}} = \epsilon + O(\epsilon^2)$ and $\leq 2\epsilon$;
- $\sqrt{1+\epsilon} = 1 + \frac{\epsilon}{2} - O(\epsilon^2)$ and $\geq 1 + \frac{\epsilon}{3}$.
Sandwiching logs. For $t > 0$, $1-\f1t \le \log t \le t-1$.
Inverting $n \log n$. If $y = x\ln x$, then $x \sim \frac{y}{\ln y}$.
Birthday paradox. Let $p_{n,m}$ be the probability that $m$ balls thrown into $n$ bins don’t collide. Then
\[p_{n,m} = \exp\p{-\frac{n^2}{m}} \underbrace{\p{1+O\p{\frac{n}{m}}}}_\text{when $m \ll \sqrt{n}$} \underbrace{\p{1-O\p{\frac{n^3}{m^2}}}}_\text{when $m \gg \sqrt{n}$}.\]
If $m = \sqrt{2 \ln 2}\sqrt{n} \pm O(1)$, then $p_{n,m} = \frac{1}{2} \pm O\p{\frac{1}{\sqrt{m}}}$.
Factorials and binomial coefficients
Stirling’s formula. $n! = \sqrt{2\pi n}\p{\frac{n}{e}}^n\p{1\pm O\p{\frac{1}{n}}}$.
Small $k$. If $k = o(\sqrt{n})$, then $\binom{n}{k} \sim \frac{n^k}{k!} \sim \frac{\p{\frac{en}{k}}^k}{\sqrt{2\pi k}}$.
General. For $1 \leq k \leq n$, $\p{\frac{n}{k}}^k \leq \binom{n}{k} \leq \frac{n^k}{k!} \leq \p{\frac{en}{k}}^k$.
Logarithm. Thus $\ln\binom{n}{k} \sim k \ln\p{\frac{n}{k}}$. For example, $\ln\binom{n}{n^{.99}} = \Theta(n^{.99}\log n)$, but $\ln\binom{n}{.01n} = \Theta(n)$.
Constant fraction. More precisely, $\binom{n}{pn} \sim \frac{1}{\sqrt{2\pi pq}}\frac{1}{\sqrt{n}}\b{\p{\frac{1}{p}}^p \p{\frac{1}{q}}^q}^n = \frac{1}{\sqrt{2\pi pq}}\frac{1}{\sqrt{n}}2^{H_2(p)n}$, where $H_2(p) \coloneqq -p\log_2p - q \log_2q$ is the “binary entropy”. In particular,
- the probability that exactly $\frac{n}{2}$ of $n$ unbiased coins turn up heads is $\binom{n}{n/2}2^{-n} \sim \sqrt{\frac{2}{\pi}} \frac{1}{\sqrt{n}}$;
- when $p$ is small (but still constant in $n$), $H_2(p) = p \log_2 (2/p) - O(p^2)$.
#to-write explain how it’s part of the more general fact about KL and histograms
Hamming volume. Let $V(n,k) \coloneqq \binom{n}{0} + \cdots + \binom{n}{k}$ be the size of a Hamming ball of radius $k$. If $k = (1/2-\Omega(1))n$, then $V(n,k) = \Theta\p{\binom{n}{k}}$. In general, if $k\le \f n 2$ then $V(n,k) \le 2^{n H(k/n)}$.
Add $w$ dummies. $\sum_{i=0}^w \binom{n}{k-i} \leq \binom{n+w}{k}$.
Go down. $\frac{\binom{n+w}{k}}{\binom{n}{k}} \leq \p{\frac{n}{n-k}}^w$.
Go right. $\frac{\binom{n}{k+w}}{\binom{n}{k}} \leq \p{\frac{n-k-w}{k}}^w \leq \p{\frac{n}{k}}^w$.
Generating functions. $\sum_{k=0}^n \binom{n}{k}x^k = (1+x)^n$, and $\sum_{n=k}^\infty \binom{n}{k}y^n = \frac{y^k}{(1-y)^{k+1}}$.
Gaussians and CLT
Let $Z \sim \Normal(0,1)$.
Density. The pdf of $Z$ is $\fi(u) \coloneqq \frac{1}{\sqrt{2\pi}}e^{-u^2/2}$.
Cumulative. For $u \geq 0$, $\p{\frac{1}{u}-\frac{1}{u^3}}\fi(u) \leq \Pr[Z \geq u] \leq \frac{1}{u}\fi(u)$. Also, $\Pr[|Z| \geq u] \leq e^{-u^2/2}$ (better for small $u$).
Berry-Esseen theorem. Let $X_1,\ldots,X_n$ be independent with $\E[X_i]=0$ and $\sum_i\Var[X_i] = 1$. Then
\[\bigl|\Pr[X_1 + \cdots + X_n \geq u] - \Pr[Z \geq u]\bigr| \leq \sum_i \E\b{|X_i|^3}.\]
In particular, if $X_1,\ldots,X_n$ are i.i.d. with mean $0$ and variance $1$ each, then
\[
\abs{\Pr\b{X_1 + \cdots + X_n \geq u\sqrt{n}} - \Pr[Z \geq u]}
\leq O\p{\frac{1}{\sqrt{n}}}.
\]
Above $u \approx \sqrt{\log n}$, $\Pr[Z \geq u]$ falls below $1/\sqrt{n}$, so you should start using Hoeffding instead.
Large-deviation bounds
Markov. If $X \geq 0$, then $\Pr[X \geq k\E[X]] \leq 1/k$.
Markov’s converse. If $0 \leq X \leq 1$, then $\Pr[X \leq \epsilon/2] \leq \epsilon/2$.
Chebyshev. Let $\mu \ce \E[X]$ and $\sigma^2 \ce \Var[X]$, then $\Pr[|X-\mu| \geq t\sigma] \leq 1/t^2$.
Hoeffding (additive). Let $X_1,\ldots,X_m$ be independent with $0 \leq X_i \leq 1$. Let $X \ce X_1+\cdots + X_n$ and $\mu \ce \E[X]$. Then
\[
\Pr\b{|X-\mu| \geq u\sqrt{n}}
\leq 2\cdot \exp\p{-2u^2},
\]
or equivalently,
\[\Pr\b{\abs{X-\mu} \geq \eps n} \leq 2\cdot \exp(-2\eps^2n).\]
In particular, the sample mean of a Bernoulli is accurate to $\pm \eps$ except with that probability. (If your variables have very different ranges/variances then the version in Hoeffding bound might be more suitable?)
Chernoff (multiplicative). Under the same assumptions
- When $\eps \in [0,1]$, $\Pr[X \not\in (1\pm\epsilon)\mu] \leq 2\cdot \exp\p{-\frac{\epsilon^2\mu}{3}}$.
- when $\eps>1$, $\Pr[X \geq (1+\eps)\mu] \leq \exp\p{-\frac{\eps\mu}{3}}$.
Chernoff when you don’t know the mean. Say all you know is $\mu \leq \mu_\mathrm{U}$, then surprisingly you can still say $\Pr[X \geq (1+\epsilon)\mu_\mathrm{U}] \leq \exp\p{-\frac{\epsilon^2\mu_\mathrm{U}}{3}}$.
Note: If your variables are “negatively correlated”, or if the aggregate function isn’t a sum but another function with nice Lipschitzness, then Chernoff-like bounds exist too.
Sampling theorem. Let $0 \leq X \leq 1$ with $\E[X]=\mu$, and let $\hat{\mu}$ be the average of $m$ i.i.d. copies of $X$. Then for any $0 < \epsilon,\delta < 1$, if $m \geq \frac{3\ln(1/\delta)}{\epsilon^2}$, then $\Pr[|\hat{\mu}- \mu| > \epsilon] \leq \delta$.
Fields and polynomials
Finite fields. The size of a finite field is either a prime (e.g. $\Z_p$) or a power of primes (e.g. quotients of $\Z_p[X]$), and they’re all the same up to isomorphism. One can construct and work in $\F_q$ in $\polylog(q)$ time.
Note: For more details on how to find them, see Lectures 10b and 10c.
Prime number theorem. The fraction of $n$-bit numbers that are prime is $\sim \frac{1}{\ln n}$ (see Density of primes for cute but untight arguments).
Finding irreducibles. If $P \in \F_q[X]$ is a uniformly random polynomial of degree $d$, then <div>[\Pr[\text{$P$ is irreducible}] \in \b{\frac{1}{2d}, \frac{1}{d}}.]</div>
Reduced polynomials. For any $x \in \F_q$, $x^q = x$. Accordingly, we say a multivariate polynomial is reduced if no variable is taken to a power greater than $q-1$. Reduced polynomials uniquely represent functions $f : \F_q^n \to \F_q$.
Schwartz-Zippel. Let $P \in \F_q[X_1,\ldots,X_n]$ be a nonzero reduced polynomial with total degree $\leq d$, and let $\alpha_1, \ldots, \alpha_n$ be drawn uniformly and independently from $S \subseteq \F_q$. Then $\Pr[P(\alpha_1, \ldots, \alpha_n) = 0] \leq \frac{d}{|S|}$. Also, if $q=2$ and $S=\F_2$, $\Pr[P(\alpha_1, \ldots, \alpha_n) = 0] \leq 1-1/2^d$.
Note: An expression for general $q$ and $d$ (e.g. $q=3$) is given in Lecture 10e.
Inequalities
$p$-distances and $p$-means. For $0<p<q$, $\norm{x}_p \geq \norm{x}_q$ but $\norm{x}_p \leq n^{\frac{1}{p}-\frac{1}{q}} \norm{x}_q$. (See p-norms.)
Bonferroni inequalities. In inclusion-exclusion, when you stop at intersections of $k$ sets, if $k$ is odd it’s an upper bound ($k=1$ is the union bound), and if $k$ is even it’s a lower bound. Equivalently, $\sum_{i=0}^k (-1)^i\binom{n}{i} \leq 0$ for $k$ odd and $\geq 0$ for $k$ even.