arXiv subjects
Publications and source records attributed to Mark Lewko.
We prove variational forms of the Barban-Davenport-Halberstam Theorem and the large sieve inequality. We apply our result to prove an estimate for the sum of the squares of prime differences, averaged over arithmetic progressions.
We investigate the square variation operator $V^2$ (which majorizes the partial sum maximal operator) on general orthonormal systems (ONS) of size $N$. We prove that the $L^2$ norm of the $V^2$ operator is bounded by $O(\ln(N))$ on any ONS. This result is sharp and refines the classical Rademacher-Menshov theorem. We show that this can be improved to $O(\sqrt{\ln(N)})$ for the trigonometric system, which is also sharp. We show that for any choice of coefficients, this truncation of the trigonometric system can be rearranged so that the $L^2$ norm of the associated $V^2$ operator is $O(\sqrt{\ln\ln(N)})$. We also show that for $p>2$, a bounded ONS of size $N$ can be rearranged so that the $L^2$ norm of the $V^p$ operator is at most $O_p(\ln \ln (N))$ uniformly for all choices of coefficients. This refines Bourgain's work on Garsia's conjecture, which is equivalent to the $V^{\infty}$ case. Several other results on operators of this form are also obtained. The proofs rely on combinatorial and probabilistic methods.
We prove certain endpoint restriction estimates for the paraboloid over finite fields in three and higher dimensions. Working in the bilinear setting, we are able to pass from estimates for characteristic functions to estimates for general functions while avoiding the extra logarithmic power of the field size which is introduced by the dyadic pigeonhole approach. This allows us to remove logarithmic factors from the estimates obtained by Mockenhaupt and Tao in three dimensions and those obtained by Iosevich and Koh in higher dimensions.
We establish an exact asymptotic formula for the square variation of certain partial sum processes. Let $\{X_{i}\}$ be a sequence of independent, identically distributed mean zero random variables with finite variance $σ$ and satisfying a moment condition $\mathbb{E}[|X_{i}|^{2+δ} ] < \infty$ for some $δ> 0$. If we let $\mathcal{P}_{N}$ denote the set of all possible partitions of the interval $[N]$ into subintervals, then we have that $\max_{π\in \mathcal{P}_{N}} \sum_{I \in π} | \sum_{i\in I} X_{i}|^2 \sim 2 σ^2N \ln \ln(N)$ holds almost surely. This can be viewed as a variational strengthening of the law of the iterated logarithm and refines results of J. Qian on partial sum and empirical processes. When $δ= 0$, we obtain a weaker `in probability' version of the result.
We investigate the structure of finite sets $A \subseteq \Z$ where $|A+A|$ is large. We present a combinatorial construction that serves as a counterexample to natural conjectures in the pursuit of an "anti-Freiman" theory in additive combinatorics. In particular, we answer a question along these lines posed by O'Bryant. Our construction also answers several questions about the nature of finite unions of $B_2[g]$ and $B^\circ_2[g]$ sets, and enables us to construct a $Λ(4)$ set which does not contain large $B_2[g]$ or $B^\circ_2[g]$ sets.
We show that the optimal constant in Erdös' sum-free subset theorem cannot be larger than $11/28 \approx .393$.