On the $L^p$ Norms of Random Exponential Sums and the Metric Domination Principle over the Simple Random Walk

Abstract

Let $R(n)$ denote the Simple Random Walk on $\mathbb{Z}$. We investigate the $L^p$-norms of random exponential sums of the form $S_A(a,y)={\sum }_{j\in A}{a}_j{e}^{2\pi iyR(j)}$ evaluated on the torus $\mathbb{T}$, where $A$ is an arbitrary finite subset of $\mathbb{N}$. By establishing a strict inductive bound on the central binomial coefficient, we precisely estimate the collision probability kernel of the random walk. To address the non-monotonicity of the walk, we introduce a Metric Domination Principle, proving that the $L^2$-energy of the random sum over any sparse set $A$ is majorized by that of a contiguous initial segment of integers of the same cardinality. Interpolating between $L^{\infty }$ and $L^2$, we show that for any $p\ge 2$, the expectation of the supremal $L^p$-norm over all bounded coefficients is universally bounded by $C_p|A{|}^{p-1/2}$.