If one replaces the real line with the Walsh ring $F_2[t](\frac{1}{t})$ (or equivalently, replaces the Fourier transform by the Fourier-Walsh transform), then Littlewood-Paley projections become precisely the same thing as martingale differences. See for instance the lecture notes of Pererya and Ward at

http://www.math.unm.edu/~crisp/papers/princeton.pdf

or my own lecture notes at

http://www.math.ucla.edu/~tao/254a.1.01w/notes5.ps

Very roughly speaking, the difference between the two is the difference between a sine wave and a square wave - and the latter, when viewed in binary, depicts the fluctuation of a random bit. In contrast, a sine wave of frequency comparable to $2^k$ (and more generally, a Littlewood-Paley projection to that range of frequencies) depends primarily, but not exclusively, of the $k^{th}$ bit in the binary expansion of the domain variable - and one can view the binary bits of the domain variable as independent random variables.

If one replaces the real line with the Walsh ring $F_2[t](\frac{1}{t})$ (or equivalently, replaces the Fourier transform by the Fourier-Walsh transform), then Littlewood-Paley projections become precisely the same thing as martingale differences. See for instance the lecture notes of Pererya and Ward at

https://web.archive.org/web/20210507042436/http://www.math.unm.edu/~crisp/papers/princeton.pdf

or my own lecture notes at

https://www.math.ucla.edu/~tao/254a.1.01w/notes5.ps

Very roughly speaking, the difference between the two is the difference between a sine wave and a square wave - and the latter, when viewed in binary, depicts the fluctuation of a random bit. In contrast, a sine wave of frequency comparable to $2^k$ (and more generally, a Littlewood-Paley projection to that range of frequencies) depends primarily, but not exclusively, of the $k^{th}$ bit in the binary expansion of the domain variable - and one can view the binary bits of the domain variable as independent random variables.

If one replaces the real line with the 2-adics (or equivalently, replaces the Fourier transform by the Fourier-Walsh transform), then Littlewood-Paley projections become precisely the same thing as martingale differences. See for instance the lecture notes of Pererya and Ward at

http://www.math.unm.edu/~crisp/papers/princeton.pdf

or my own lecture notes at

http://www.math.ucla.edu/~tao/254a.1.01w/notes5.ps

Very roughly speaking, the difference between the two is the difference between a sine wave and a square wave - and the latter, when viewed in binary, depicts the fluctuation of a random bit.

If one replaces the real line with the Walsh ring $F_2[t](\frac{1}{t})$ (or equivalently, replaces the Fourier transform by the Fourier-Walsh transform), then Littlewood-Paley projections become precisely the same thing as martingale differences. See for instance the lecture notes of Pererya and Ward at

http://www.math.unm.edu/~crisp/papers/princeton.pdf

or my own lecture notes at

http://www.math.ucla.edu/~tao/254a.1.01w/notes5.ps

Very roughly speaking, the difference between the two is the difference between a sine wave and a square wave - and the latter, when viewed in binary, depicts the fluctuation of a random bit. In contrast, a sine wave of frequency comparable to $2^k$ (and more generally, a Littlewood-Paley projection to that range of frequencies) depends primarily, but not exclusively, of the $k^{th}$ bit in the binary expansion of the domain variable - and one can view the binary bits of the domain variable as independent random variables.