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Let $f(z) = \sum_{n=1}^\infty a_f(n)q^n$ be a holomorphic cuspidal newform with even integral weight $k\geq 2$, level $N$, trivial nebentypus, and no complex multiplication (CM). For all primes $p$, we may define $\theta_p\in [0,\pi]$ such that $a_f(p) = 2p^{(k-1)/2}\cos \theta_p$. The Sato-Tate conjecture states that the angles $\theta_p$ are equidistributed with respect to the probability measure $\mu_{\textrm{ST}}(I) = \frac{2}{\pi}\int_I \sin^2 \theta \; d\theta$, where $I\subseteq [0,\pi]$. Using recent results on the automorphy of symmetric-power $L$-functions due to Newton and Thorne, we explicitly bound the error term in the Sato-Tate conjecture when $f$ corresponds to an elliptic curve over $\mathbb{Q}$ of arbitrary conductor or when $f$ has squarefree level. In these cases, if $\pi_{f,I}(x) := \#\{ p \leq x : p \nmid N, \theta_p\in I\}$, and $\pi(x) := \# \{ p \leq x \}$, we prove the following bound: $$\left| \frac{\pi_{f,I}(x)}{\pi(x)} - \mu_{\textrm{ST}}(I)\right| \leq 58.1\frac{\log((k-1)N \log{x})}{\sqrt{\log{x}}} \qquad \text{for} \quad x \geq 3.$$ As an application, we give an explicit bound for the number of primes up to $x$ that violate the Atkin-Serre conjecture for $f$., Comment: 31 pages; minor improvements in exposition; minor typos fixed

A positive linear recurrence sequence (PLRS) is a sequence defined by a homogeneous linear recurrence relation with positive coefficients and a particular set of initial conditions. A sequence of positive integers is \emph{complete} if every positive integer is a sum of distinct terms of the sequence. One consequence of Zeckendorf's theorem is that the sequence of Fibonacci numbers is complete. Previous work has established a generalized Zeckendorf's theorem for all PLRS's. We consider PLRS's and want to classify them as complete or not. We study how completeness is affected by modifying the recurrence coefficients of a PLRS. Then, we determine in many cases which sequences generated by coefficients of the forms $[1, \ldots, 1, 0, \ldots, 0, N]$ are complete. Further, we conjecture bounds for other maximal last coefficients in complete sequences in other families of PLRS's. Our primary method is applying Brown's criterion, which says that an increasing sequence $\{H_n\}_{n = 1}^{\infty}$ is complete if and only if $H_1 = 1$ and $H_{n + 1} \leq 1 + \sum_{i = 1}^n H_i$. This paper is an introduction to the topic that is explored further in Completeness of Positive Linear Recurrence Sequences arXiv:2010.01655., Comment: arXiv admin note: substantial text overlap with arXiv:2010.01655

A sequence of positive integers is complete if every positive integer is a sum of distinct terms. A positive linear recurrence sequence (PLRS) is a sequence defined by a homogeneous linear recurrence relation with nonnegative coefficients of the form $H_{n+1} = c_1 H_n + \cdots + c_L H_{n-L+1}$ and a particular set of initial conditions. We seek to classify various PLRS's by completeness. With results on how completeness is affected by modifying the recurrence coefficients of a PLRS, we completely characterize completeness of several families of PLRS's as well as conjecturing criteria for more general families. Our primary method is applying Brown's criterion, which says that an increasing sequence $\{H_n\}_{n = 1}^{\infty}$ is complete if and only if $H_1 = 1$ and $H_{n + 1} \leq 1 + \sum_{i = 1}^n H_i$. %A survey of these results can be found in \cite{BHLLMT}. Finally, we adopt previous analytic work on PLRS's to find a more efficient way to check completeness. Specifically, the characteristic polynomial of any PLRS has exactly one positive root; by bounding the size of this root, the majority of sequences may be classified as complete or incomplete. Additionally, we show there exists an indeterminate region where the principal root does not reveal any information on completeness. We have conjectured precise bounds for this region., Comment: 45 pages, 1 figure

The Erd\H{o}s distance problem concerns the least number of distinct distances that can be determined by $N$ points in the plane. The integer lattice with $N$ points is known as \textit{near-optimal}, as it spans $\Theta(N/\sqrt{\log(N)})$ distinct distances, the lower bound for a set of $N$ points (Erd\H{o}s, 1946). The only previous non-asymptotic work related to the Erd\H{o}s distance problem that has been done was for $N \leq 13$. We take a new non-asymptotic approach to this problem in a model case, studying the distance distribution, or in other words, the plot of frequencies of each distance of the $N\times N$ integer lattice. In order to fully characterize this distribution, we adapt previous number-theoretic results from Fermat and Erd\H{o}s in order to relate the frequency of a given distance on the lattice to the sum-of-squares formula. We study the distance distributions of all the lattice's possible subsets; although this is a restricted case, the structure of the integer lattice allows for the existence of subsets which can be chosen so that their distance distributions have certain properties, such as emulating the distribution of randomly distributed sets of points for certain small subsets, or emulating that of the larger lattice itself. We define an error which compares the distance distribution of a subset with that of the full lattice. The structure of the integer lattice allows us to take subsets with certain geometric properties in order to maximize error; we show these geometric constructions explicitly. Further, we calculate explicit upper bounds for the error when the number of points in the subset is $4$, $5$, $9$ or $\left \lceil N^2/2\right\rceil$ and prove a lower bound in cases with a small number of points., Comment: 18 pages