The following seven arithmetic lemmas D1 to D7 (the ‘D’ refers to the surface-to-volume ratio that appears in some of them) can be found in section 4.1 of Arnold (1963a). This is a central section, but Arnold is quite brief there. He derives in these lemmas the measure of the frequency domain that remains after the removal of boundary layers and interior strips from a domain Ω. These removals are required for estimates of Fourier coefficients from complex function theory and in order to avoid resonances.