The main purpose of this thesis is to derive an upper bound and a lower bound in a law of the iterated logarithm for sums of the form k=1N akf(nkx + ck) where the nk satisfy a Hadamard gap condition and ck ∈ Rn . Here we assume that f is a Dini continuous function on Rn which satisfies the property that for every cube Q of sidelength 1 with corners in the lattice Zn , f vanishes on ∂Q and has mean value zero on Q. And for the lower bound result, we need an extra condition that f has the property that there exists a number c0 > 0 such that 1Q Q |f(u)|2du > c0 for all cubes of sidelength at least 1, so that we can keep f from becoming too "sparse" at infinity. We will introduce an important concept, dyadic martingales, and then proof of our theorems can be obtained by using a reduction to dyadic martingales.
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