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Roger C. Baker

Publications and source records attributed to Roger C. Baker.

8 recordsLinked to original sources
arXiv2021

Additive energy and a large sieve inequality for sparse sequences

We consider the large sieve inequality for sparse sequences of moduli and give a general result depending on the additive energy (both symmetric and asymmetric) of the sequence of moduli. For example, in the case of monomials $f(X) = X^k$ this allows us to improve, in some ranges of the parameters, the previous bounds of S. Baier and L. Zhao (2005), K.~Halupczok (2012, 2015, 2018) and M.~Munsch (2020). We also consider moduli defined by polynomials $f(X) \in \mathbb{Z}[X]$, Piatetski-Shapiro sequences and general convex sequences. We then apply our results to obtain a version of the Bombieri--Vinogradov theorem with Piatetski-Shapiro moduli improving the level of distribution of R.~C.~Baker (2014).

math.NT↗
arXiv2021

Large Weyl sums and Hausdorff dimension

We obtain the exact value of the Hausdorff dimension of the set of coefficients of Gauss sums which for a given $α\in (1/2,1)$ achieve the order at least $N^α$ for infinitely many sum lengths $N$. For Weyl sums with polynomials of degree $d\ge 3$ we obtain a new upper bound on the Hausdorff dimension of the set of polynomial coefficients corresponding to large values of Weyl sums. Our methods also work for monomial sums, match the previously known lower bounds, just giving exact value for the corresponding Hausdorff dimension when $α$ is close to $1$. We also obtain a nearly tight bound in a similar question with arbitrary integer sequences of polynomial growth.

math.NT↗
arXiv2021

Bounds on the Norms of Maximal Operators on Weyl Sums

We obtain new estimates on the maximal operator applied to the Weyl sums. We also consider the quadratic case (that is, Gauss sums) in more details. In wide ranges of parameters our estimates are optimal and match lower bounds. Our approach is based on a combination of ideas of Baker (2021) and Chen and Shparlinski (2020).

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arXiv2016

Gaps of Smallest Possible Order between Primes in an Arithmetic Progression

Let $t \in \mathbb{N}$, $η>0$. Suppose that $x$ is a sufficiently large real number and $q$ is a natural number with $q \leq x^{5/12-η}$, $q$ not a multiple of the conductor of the exceptional character $χ^*$ (if it exists). Suppose further that, \[ \max \{p : p | q \} < \exp (\frac{\log x}{C \log \log x}) \; \; {and} \; \; \prod_{p | q} p < x^δ, \] where $C$ and $δ$ are suitable positive constants depending on $t$ and $η$. Let $a \in \mathbb{Z}$, $(a,q)=1$ and \[ \mathcal{A} = \{n \in (x/2, x]: n \equiv a \pmod{q} \} . \] We prove that there are primes $p_1 < p_2 < ... < p_t$ in $\mathcal{A}$ with \[ p_t - p_1 \ll qt \exp (\frac{40 t}{9-20 θ}) . \] Here $θ= (\log q) / \log x$.

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arXiv2015

Clusters of primes with square-free translates

Let $\mathcal{R}$ be a finite set of integers satisfying appropriate local conditions. We show the existence of long clusters of primes $p$ in bounded length intervals with $p-b$ squarefree for all $b \in \mathcal{R}$. Moreover, we can enforce that the primes $p$ in our cluster satisfy any one of the following conditions: (1) $p$ lies in a short interval $[N, N+N^{\frac{7}{12}+ε}]$, (2) $p$ belongs to a given inhomogeneous Beatty sequence, (3) with $c \in (\frac{8}{9},1)$ fixed, $p^c$ lies in a prescribed interval mod $1$ of length $p^{-1+c+ε}$.

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arXiv2014

Bounded gaps between primes with a given primitive root, II

Let $m$ be a natural number, and let $\mathcal{Q}$ be a set containing at least $\exp(C m)$ primes. We show that one can find infinitely many strings of $m$ consecutive primes each of which has some $q\in\mathcal{Q}$ as a primitive root, all lying in an interval of length $O_{\mathcal{Q}}(\exp(C'm))$. This is a bounded gaps variant of a theorem of Gupta and Ram Murty. We also prove a result on an elliptic analogue of Artin's conjecture. Let $E/\mathbb{Q}$ be an elliptic curve with an irrational $2$-torsion point. Assume GRH. Then for every $m$, there are infinitely many strings of $m$ consecutive primes $p$ for which $E(\mathbb{F}_p)$ is cyclic, all lying an interval of length $O_E(\exp(C'' m))$. If $E$ has CM, then the GRH assumption can be removed. Here $C$, $C'$, and $C''$ are absolute constants.

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arXiv2012

Piatetski-Shapiro sequences

We consider various arithmetic questions for the Piatetski-Shapiro sequences $\fl{n^c}$ ($n=1,2,3,...$) with $c>1$, $c\not\in\N$. We exhibit a positive function $θ(c)$ with the property that the largest prime factor of $\fl{n^c}$ exceeds $n^{θ(c)-\eps}$ infinitely often. For $c\in(1,\tfrac{149}{87})$ we show that the counting function of natural numbers $n\le x$ for which $\fl{n^c}$ is squarefree satisfies the expected asymptotic formula. For $c\in(1,\tfrac{147}{145})$ we show that there are infinitely many Carmichael numbers composed entirely of primes of the form $p=\fl{n^c}$.

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