arXiv subjects

Sam Chow

Publications and source records attributed to Sam Chow.

46 records · Page 3Linked to original sources

arXiv2017

Bohr sets and multiplicative diophantine approximation

In two dimensions, Gallagher's theorem is a strengthening of the Littlewood conjecture that holds for almost all pairs of real numbers. We prove an inhomogeneous fibre version of Gallagher's theorem, sharpening and making unconditional a result recently obtained conditionally by Beresnevich, Haynes and Velani. The idea is to find large generalised arithmetic progressions within inhomogeneous Bohr sets, extending a construction given by Tao. This precise structure enables us to verify the hypotheses of the Duffin--Schaeffer theorem for the problem at hand, via the geometry of numbers.

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arXiv2016

A note on rational points near planar curves

Under fairly natural assumptions, Huang counted the number of rational points lying close to an arc of a planar curve. He obtained upper and lower bounds of the correct order of magnitude, and conjectured an asymptotic formula. In this note, we establish the conjectured asymptotic formula.

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arXiv2015

Equidistribution of values of linear forms on a cubic hypersurface

Let $C$ be a cubic form with rational coefficients in $n$ variables, and let $h$ be the $h$-invariant of $C$. Let $L_1, \ldots, L_r$ be linear forms with real coefficients such that if $\boldsymbolα \in \mathbb{R}^r \setminus \{ \boldsymbol{0} \}$ then $\boldsymbolα \cdot \mathbf{L}$ is not a rational form. Assume that $h > 16 + 8 r$. Let $\boldsymbolτ \in \mathbb{R}^r$, and let $η$ be a positive real number. We prove an asymptotic formula for the weighted number of integer solutions $\mathbf{x} \in [-P,P]^n$ to the system $C(\mathbf{x}) = 0, \: |\mathbf{L}(\mathbf{x}) - \boldsymbolτ| < η$. If the coefficients of the linear forms are algebraically independent over the rationals, then we may replace the $h$-invariant condition with the hypothesis $n > 16 + 9 r$, and show that the system has an integer solution. Finally, we show that the values of $\mathbf{L}$ at integer zeros of $C$ are equidistributed modulo one in $\mathbb{R}^r$, requiring only that $h > 16$.

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arXiv2014

Rationality and power

We produce an infinite family of transcendental numbers which, when raised to their own power, become rational. We extend the method, to investigate positive rational solutions to the equation $x^x = α$, where $α$ is a fixed algebraic number. We then explore the consequences of $x^{P(x)}$ being rational, if $x$ is rational and $P(x)$ is a fixed integer polynomial.

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arXiv2014

Sums of cubes with shifts

Let $μ_1, \ldots, μ_s$ be real numbers, with $μ_1$ irrational. We investigate sums of shifted cubes $F(x_1,\ldots,x_s) = (x_1 - μ_1)^3 + \ldots + (x_s - μ_s)^3$. We show that if $η$ is real, $τ>0$ is sufficiently large, and $s \ge 9$, then there exist integers $x_1 > μ_1, \ldots, x_s > μ_s$ such that $|F(\mathbf{x})- τ| < η$. This is a real analogue to Waring's problem. We then prove a full density result of the same flavour for $s \ge 5$. For $s \ge 11$, we provide an asymptotic formula. If $s \ge 6$ then $F(\mathbf{Z}^s)$ is dense on the reals. Given nine variables, we can generalise this to sums of univariate cubic polynomials.

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arXiv2014

Waring's problem with shifts

Let $μ_1, \ldots, μ_s$ be real numbers, with $μ_1$ irrational. We investigate sums of shifted $k$th powers $\mathfrak{F}(x_1, \ldots, x_s) = (x_1 - μ_1)^k + \ldots + (x_s - μ_s)^k$. For $k \ge 4$, we bound the number of variables needed to ensure that if $η$ is real and $τ> 0$ is sufficiently large then there exist integers $x_1 > μ_1, \ldots, x_s > μ_s$ such that $|\mathfrak{F}(\mathbf{x}) - τ| < η$. This is a real analogue to Waring's problem. When $s \ge 2k^2-2k+3$, we provide an asymptotic formula. We prove similar results for sums of general univariate degree $k$ polynomials.

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arXiv2014

Distinguishing newforms

Let $n_0(N,k)$ be the number of initial Fourier coefficients necessary to distinguish newforms of level $N$ and even weight $k$. We produce extensive data to support our conjecture that if $N$ is a fixed squarefree positive integer and $k$ is large then $n_0(N,k)$ is the least prime that does not divide $N$.

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arXiv2013

Averaging on thin sets of diagonal forms

We investigate one-dimensional families of diagonal forms, considering the evolution of the asymptotic formula and error term. We then discuss properties of the average asymptotic formula obtained. The subsequent second moment analysis precipitates an effective means of computing $p$-adic densities of zeros for large primes $p$.

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arXiv2013

Cubic diophantine inequalities for split forms

Denote by $s_0^{(r)}$ the least integer such that if $s \ge s_0^{(r)}$, and $F$ is a cubic form with real coefficients in $s$ variables that splits into $r$ parts, then $F$ takes arbitrarily small values at nonzero integral points. We bound $s_0^{(r)}$ for $r \le 6$.

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arXiv2013

Distinguishing eigenforms modulo a prime ideal

Consider the Fourier expansions of two elements of a given space of modular forms. How many leading coefficients must agree in order to guarantee that the two expansions are the same? Sturm gave an upper bound for modular forms of a given weight and level. This was adapted by Ram Murty, Kohnen and Ghitza to the case of two eigenforms of the same level but having potentially different weights. We consider their expansions modulo a prime ideal, presenting a new bound. In the process of analysing this bound, we generalise a result of Bach and Sorenson, who provide a practical upper bound for the least prime in an arithmetic progression.

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