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Theophilus Agama

Publications and source records attributed to Theophilus Agama.

At least 19 recordsLinked to original sources
arXiv2026

On the Erdős distance problem

In this paper, using the compression method, we recover the lower bound for the Erdős unit distance problem and provide an alternative proof to the distinct distance conjecture. In particular, in $\mathbb{R}^k$ for all $k\geq 2$, we have \begin{align} \#\bigg\{(\vec{x}_t,\vec{x_j})\in \mathbb{E}\subset\mathbb{R}^k~:~||\vec{x_j}-\vec{x_t}||=1,~1\leq t,j\leq n\bigg\}\geq C\frac{\sqrt{k}}{2}n^{1+o(1)}\nonumber \end{align} for some $C>0$. We also show that \begin{align} \# \bigg\{d_j:d_j=||\vec{x_s}-\vec{y_t}||,~d_j\neq d_i,~1\leq s,t\leq n\bigg\}\geq D\frac{\sqrt{k}}{2}n^{\frac{2}{k}-o(1)}\nonumber \end{align} for some $D>0$. These lower bounds generalize the lower bounds of the Erdős unit distance and the distinct distance problem to higher dimensions.

math.MG↗
arXiv2026

New bounds for the Heilbronn triangle problem

Using ideas from the geometry of compression, we improve on the current upper and lower bounds of the Heilbronn triangle problem. In particular, let $Δ(s)$ denote the minimal area of the triangle induced by $s$ points on a unit disk. We have the upper bound $$ Δ(s)\ll \frac{1}{s^{\frac{3}{2}-ε}} $$ for small $ε:=ε(s)>0$ and the lower bound $$ Δ(s)\gg \frac{\log s}{s\sqrt{s}}. $$

math.NT↗
arXiv2026

Complex Circles of Partition and the Expansion Principles

In this paper, we further develop the theory of circles of partition by introducing the notion of complex circles of partition. This work generalizes the classical framework, extending from subsets of the natural numbers as base sets to partitions defined within the complex plane, which now serves as both the base and bearing set. We employ the squeeze principle as a central tool for rigorously investigating the possibility to partition numbers with base set as a certain subset of the complex plane.

math.GM↗
arXiv2026

On the distribution of addition chains

Addition chains are a classical construction for fast exponentiation and related computation problems. In this paper, we study a chain for a fixed integer $n$ by decomposing each generator into a \emph{determiner} and a \emph{regulator} (gap). This viewpoint leads to explicit identities for two aggregate statistics of the chain: the sum of the determiners and the sum of the chain elements. We then derive the corresponding lower bounds by using the positivity of the regulators. In parallel, we establish an identity for the reciprocal sum of the chain, showing how the harmonic profile of the chain can also be written in terms of the same gap sequence. These identities provide a unified way to compare addition chains of the same target and length. The paper concludes with a balancing problem that asks for the chain(s) that minimize the difference between the arithmetic sum and the harmonic sum, together with a structural decomposition of that optimization objective.

math.NT↗
arXiv2026

The Asymptotic Binary Goldbach and Lemoine Conjectures

In this paper, we use the former of the authors developed theory of \emph{circles of partition} to investigate possibilities to prove the binary Goldbach and Lemoine conjectures. We state the \emph{squeeze principle} and its consequences when the set of all odd prime numbers is the base set. Using this tool, we can prove asymptotic versions of the binary Goldbach and the Lemoine conjecture.

math.NT↗
arXiv2026

On the pointwise periodicity of multiplicative and additive functions

We study the problem of estimating the number of points of coincidences of an idealized gap on the set of integers under a given multiplicative function $g:\mathbb{N}\longrightarrow \mathbb{C}$ respectively additive function $f:\mathbb{N}\longrightarrow \mathbb{C}$. We obtain various lower bounds depending on the length of the period, by varying the worst growth rates of the ratios of their consecutive values.

math.NT↗
arXiv2026

Studies in Additive Number Theory by Circles of Partition

In this paper, we introduce and develop the circle embedding method. This method hinges essentially on a combinatorial-geometric structure which we choose to call circles of partition. We provide applications in the context of problems that relates to deciding on the feasibility of partitioning numbers into certain subset of integers. In particular, our method allows us to partition any sufficiently large number $n\in\mathbb{N}$ into any set $\mathbb{H}$ with natural density strictly greater than $\frac{1}{2}$. This possibility could herald an unprecedented progress on categories of problems of similar flavour. The paper finishes by presenting an asymptotic proof of the binary Goldbach and Lemoine conjecture as an application of the developed method.

math.GM↗
arXiv2026

On the general no-three-in-line problem

In this paper, we show that the number of points that can be placed in the grid $n\times n\times \cdots \times n~(d~times)=n^d$ for all $d\in \mathbb{N}$ with $d\geq 2$ so that no three points are collinear satisfies the lower bound \begin{align} \gg n^{d-1}\sqrt[2d]{d}.\nonumber \end{align} This extends the result of the no-three-in-line problem to all dimension $d\geq 3$.

math.CO↗
arXiv2026

On the flint hills series

In this note, we study the flint hills series of the form \begin{align} \sum \limits_{n=1}^{\infty}\frac{1}{(\sin^2n) n^3}\nonumber \end{align} via a certain method. The method essentially works by erecting certain pillars sufficiently close to the terms in the series and evaluating the series at those spots. This allows us to relate the convergence and the divergence of the series to other series that are somewhat tractable. In particular, we show that the convergence of the flint hill series relies very heavily on the condition that for any small $ε>0$ \begin{align} \bigg|\sum \limits_{i=0}^{\frac{n+1}{2}}\sum \limits_{j=0}^{i}(-1)^{i-j}\binom{n}{2i+1} \binom{i}{j}\bigg|^{2s} \leq |(\sin^2n)|n^{2s+2-ε}\nonumber \end{align} for some $s\in \mathbb{N}$.

math.GM↗
arXiv2026

On the Gap sequence and the Gilbreath conjecture

Motivated by the Gilbreath conjecture, we develop the notion of the gap sequence induced by any sequence of numbers. We introduce the notion of the path and associated circuits induced by an originator and study the conjecture via the notion of the trace and length of a path.

math.CO↗
arXiv2026

On addition chains and progress on the Scholz conjecture

In this paper, we develop some new classes of methods to study the Scholz conjecture on addition chains. It turns out that the exponents of numbers of the form $2^n-1$ largely determine the length of the shortest addition chain for the number that leads to $2^n-1$. Using the carry analysis, we obtain improved upper bounds for the length of the shortest addition chains $\ell(2^n-1)$ producing $2^n-1$. In particular, we show that if $2^n-1$ has carry of degree at most $$ κ(2^n-1)=\frac{1}{2}\left(\ell(n)-\left\lfloor\frac{\log n}{\log 2}\right\rfloor+\sum \limits_{j=1}^{\lfloor \frac{\log n}{\log 2}\rfloor}\left\{\frac{n}{2^j}\right\}\right) $$ then $$ \ell(2^n-1)\leq n+1+\sum \limits_{j=1}^{\lfloor\frac{\log n}{\log 2}\rfloor}\bigg(\left\{\frac{n}{2^j}\right\}-ξ(n,j)\bigg)+\ell(n) $$ for all $n\in \mathbb{N}$ with $n\geq 4$, where $\ell(\cdot)$ denotes the length of the shortest addition chain that leads to $\cdot$, $\{\cdot\}$ denotes the fractional part of $\cdot$ and where $ξ(n,1):=\{\frac{n}{2}\}$ with $ξ(n,2)=\{\frac{1}{2}\lfloor \frac{n}{2}\rfloor\}$ and so on.

math.GM↗
arXiv2026

The Compression method and applications

In this paper, we introduce and develop the method of compression of points in space. We introduce the notion of the mass, the rank, the entropy, the cover and the energy of compression. We leverage this method to prove some class of inequalities related to Diophantine equations. In particular, we show that for each $L n-1$, there exist some $(x_1,x_2,\ldots,x_n)\in \mathbb{N}^n$ with $x_i\neq x_j$ for all $1\leq i n-1$ there exist some $(x_1,x_2,\ldots,x_n)$ with $x_i\neq x_j$ for all $1\leq i<j\leq n$ and some $s\geq 2$ such that \begin{align} \sum \limits_{j=1}^{n}\frac{1}{x_j^s}\gg s\frac{n}{L^{s-1}}.\nonumber \end{align}

math.NT↗
arXiv2026

The area method and applications

In this paper, we develop a general method for estimating correlations of the forms \begin{align} \sum \limits_{n\leq x}G(n)G(x-n)\nonumber \end{align} and \begin{align} \sum \limits_{n\leq x}G(n)G(n+l)\nonumber \end{align} for a fixed $1\leq l\leq x$ and where $G:\mathbb{N}\longrightarrow \mathbb{R}^{+}$. To distinguish between the two types of correlation, we call the first correlation the \textbf{type} $2$ correlation and the second the \textbf{type} $1$ correlation. As an application, we estimate the lower bound for the \textbf{type} $2$ correlation of the master function \begin{align} \sum \limits_{n\leq x}Υ(n)Υ(n+l_0)\geq (1+o(1))\frac{x}{2\mathcal{C}(l_0)}\log \log ^2x\nonumber \end{align} provided that $Υ(n)Υ(n+l_0)>0$. We also use this method to provide a first proof of the twin prime conjecture showing that \begin{align} \sum\limits_{n\leq x}Λ(n)Λ(n+2)\geq (1+o(1))\frac{x}{2\mathcal{C}(2)}\nonumber \end{align} for some $\mathcal{C}:=\mathcal{C}(2)>0$.

math.GM↗
arXiv2026

Multivariate expansivity theory and Pierce-Birkhoff conjecture

Motivated by the Pierce-Birkhoff conjecture, we launch an extension program for single variable expansivity theory. We study this notion under tuples of polynomials in the ring $\mathbb{R}[x_1,x_2,\ldots,x_n]$. As an application, we develop some class of inequalities to study the Pierce-Birkhoff conjecture.

math.CA↗
arXiv2026

Expansivity theory and Sendov's conjecture

In this paper we introduce and develop the concept of expansivity of a tuple whose entries are elements from the polynomial ring $\mathbb{C}[x]$. As an inverse problem, we examine how to recover a tuple from the expanded tuple at any given phase of expansion. We convert the celebrated Sendov conjecture concerning the distribution of zeros of polynomials and their critical points into this language and prove some weak variants of this conjecture. We also apply this to the existence of solutions to differential equations. In particular, we show that a certain system of differential equation has no non-trivial solution. As an application we give a proof of Sendov's conjecture. We start by establishing the uniformly diminishing state of the mass of an expansion.

math.RA↗
arXiv2026

A progress on the binary Goldbach conjecture

In this paper, we develop the method of circle of partitions and associated statistics. As an application we prove conditionally the binary Goldbach conjecture. We develop a series of steps to prove the binary Goldbach conjecture in full. We end the paper by proving the binary Goldbach conjecture for all sufficiently large even numbers.

math.NT↗