Ernest S. Croot
We answer several questions of P. Erdos and R. Graham by showing that for any rational number r > 0, there exist integers n1, n2, ..., nk, k variable, where N < n1 < n2 < ... < nk < (e^r + o_r(1) ) N, such that r = 1/n1 + 1/n2 + ... + 1/nk. Moreover, we obtain the best-possible error term for the o_r(1), which is O_r(loglog N/log N).
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Rational numbers with small denominators in short intervals
本文利用Kloosterman分数的双线性型估计，改进了Balazard和Martin（2023）关于短区间内有理数最小分母平均值渐近公式的误差项。
We compute the probability mass function of the random variable which returns the smallest denominator of a reduced fraction in a randomly chosen real interval of radius $δ/2$. As an application, we prove that the expected value of the smallest denominator is asymptotic, as $δ\rightarrow 0$, to $(16/π^2)δ^{-1/2}.$
Unit fractions with shifted prime denominators
该研究证明了任何正有理数均可表示为具有特定移位质数分母的不同单位分数之和，并探讨了同余条件下的扩展情况。
An algorithm for Egyptian fraction representations with restricted denominators
本文提出了一种算法，利用给定正整数有限多重集作为分母，寻找给定有理数的所有单位分数表示，解决了传统算法分母过大且无法限制分母的问题，特别适合计算具有预设最大分母的密集埃及分数表示。
In this paper we obtain a sharp upper bound for the number of solutions to a certain diophantine inequality involving fractions with power denominator. This problem is motivated by a conjecture of Zhao concerning the spacing of such fractions in short intervals and the large sieve for power modulus. As applications of our estimate we show Zhao's conjecture is true except for a set of small measure and give a new $\ell_1 \rightarrow \ell_2$ large sieve inequality for power modulus.
Resolving a conjecture of Zhi-Wei Sun, we prove that every rational number can be represented as a sum of distinct unit fractions whose denominators are practical numbers. The same method applies to allowed denominators that are closed under multiplication by two and include a multiple of every positive integer, including the odious numbers, evil numbers, Hardy-Ramanujan numbers, Jordan-Polya numbers, and fibbinary numbers.
The average value of the smallest denominator of a rational number belonging to the interval $](j-1)/N,j/N]$, where~$j=1,\dots, N$, is proved to be asymptotically equivalent to~$16π^{-2}\sqrt{N}$, when $N$ tends to infinity. The result had been conjectured in 1977 by Kruyswijk and Meijer.
Let D(n) be the set of all fractions in the unit interval whose denominator in lowest terms equals $n$. We evaluate the product of the values of the Gamma function at the points of D(n), as a function of $n$; the answer depends on whether or not $n$ is a prime power.
We study some subsets of rational points in an algebraic groups defined by open conditions on their projection in the finite adeles points. Using adelic mixing we are able to prove an equidistribution's result for the projection of these sets in the real points. As an application, we study the existence and the repartition of rational unitary matrices having a given denominator. We prove a local-global principle for this problem and the equirepartition of the sets of denominator n-matrices when they are not empty. Then we study the more complicated case of non simply-connected groups applying …
The log moments of smallest denominators
本文研究随机中心收缩区间内有理数最小分母的对数矩，推导一维极限分布矩的显式公式，解答了环面映射最小共振阶相关研究中提出的问题。
Rational numbers with odd greedy expansion of fixed length
研究奇贪婪展开的有限性问题，找到了展开长度为2的既约分数、给定前缀长度为m的有理数及无穷族有理数。
The smallest denominator not contained in a unit fraction decomposition of $1$ with fixed length
本文研究将1表示为k个不同单位分数之和时，未出现在分母中的最小整数v(k)，并证明存在常数c>0使得v(k) ≥ e^{c k^2}，改进了Erdős和Graham提出的下界v(k) ≫ k!。
Our purpose is to give an account of the $r$-tuple problem on the increasing sequence of reduced fractions having denominators bounded by a certain size and belonging to a fixed real interval. We show that when the size grows to infinity, the proportion of the $r$-tuples of consecutive denominators with components in certain apriori fixed arithmetic progressions with the same ratio approaches a limit, which is independent on the interval. The limit is given explicitly and it is completely described in a few particular instances.
Perfect powers in the product of denominators of elliptic curves
利用筛法论证估计有理点s元组中分母乘积为固定素数ℓ次完美幂的频率，并推广到有理函数情形。
An Egyptian fraction is a sum of distinct unit fractions (reciprocals of positive integers). We show that every rational number has Egyptian fraction representations where the number of terms is of the same order of magnitude as the largest denominator, improving a result from an earlier paper to best-possible form. We also settle, in best-possible form, the related problem of how small M_t(r) can be such that there is an Egyptian fraction representation of r with exactly t terms, the denominators of which are all at most M_t(r). We also consider the following problems posed by Erdos and Graha…
A Divisor Parametrization for the Erdős--Straus Conjecture
本文引入基于除数的函数 fab(n,a,b) 来参数化所有分母被 4 整除的 1/n 的三项埃及分数分解，并与已知的 Type I/II 描述进行比较。
On the rate of convergence of continued fraction statistics of random rationals
本文利用改进的 $SL_2(\mathbb{R})/SL_2(\mathbb{Z})$ 空间上测度不逃逸与等分布结论，证明了单位区间内大分母随机有理数连分数统计量以多项式速度逼近 Gauss-Kuzmin 统计量，并推广至分子为素数的情形。
Unit fractions with semiprime denominators: an elementary proof of Erdős Problem #306
本文给出一个初等证明，证明每个分母无平方因数的正有理数可表示为不同单位分数（分母为两个不同素数乘积）的有限和，解决了 Erdős 问题 #306，并已在 Lean 4 中形式化。
Let $ p_{i, n}, i=1,..., N(n)=2^{n-1} $ be the lengths of intervals between the neighboring fractions of Brocot sequence $ F_n $. We obtain an asymptotic formula for $σ_β(F_n)=\sum_{i=1}^{N(n)} p_{i, n}^β $,which improves known estimates.
One example of singular representations of real numbers from the unit interval
本文推广文献[25]的算子方法以建模数字系统，并引入与探讨此类系统的一个例子，为单位区间实数奇异表示提供具体实例。
It is known that there is a constant $c > 0$ such that for every sequence $x_1, x_2, \ldots$ in $[0,1)$ we have for the star discrepancy $D_N^*$ of the first $N$ elements of the sequence that $N D_N^* \ge c \cdot \log N$ holds for infinitely many $N$. Let $c^*$ be the supremum of all such $c$ with this property. We show $c^* > 0.0646363$, thereby improving the until now known estimates.
It is known that there is a constant $c>0$ such that for every sequence $x_1, x_2,\ldots$ in $[0,1)$ we have for the star discrepancy $D^{*}_N$ of the first $N$ elements of the sequence that $N D^{*}_N\geq c\cdot \log N$ holds for infinitely many $N$. Let $c^{*}$ be the supremum of all such $c$ with this property. We show $c^{*}>0.065664679\ldots$, thereby slightly improving the estimates known until now.
The Farey sequence is the sequence of all rational numbers in the real unit interval, stratified by increasing denominators. A classical result by Hall says that its normalized gap distribution is the same as the distribution of the random variable 1/(2 zeta(2) xy) on a certain unit triangle. In this paper we weight the denominators by an arbitrary piecewise-smooth continuous function, and we characterize the resulting gap distribution as that of a multiple of the above variable, defined on a certain unit pentagon. Our characterization refines previous results by Boca, Cobeli and Zaharescu, bu…
Eventually greedy best Egyptian underapproximations of rational numbers via optimal control
该研究将埃及下逼近问题转化为最优控制问题，证明正有理数存在最终贪心最优埃及下逼近，还构造了具有唯一此类逼近的无理数，解答了Erdős、Graham及Nathanson的相关问题。
For fixed $t\ge 2$, we consider the class of representations of $1$ as sum of unit fractions whose denominators are powers of $t$ or equivalently the class of canonical compact $t$-ary Huffman codes or equivalently rooted $t$-ary plane "canonical" trees. We study the probabilistic behaviour of the height (limit distribution is shown to be normal), the number of distinct summands (normal distribution), the path length (normal distribution), the width (main term of the expectation and concentration property) and the number of leaves at maximum distance from the root (discrete distribution).
A simple inequality relating the Euler-Riemann zeta function, digamma, and cotangent over the unit interval
本文证明了一个与欧拉-黎曼ζ函数、digamma函数和余切函数相关的简单不等式，并提出猜想性加强及问题起源的评论。
Cyclotomic factors and irreducibility of the denominators of $q$-deformed rational numbers
本文研究 $q$-变形有理数分母的分圆因子整除性，证明相关猜想，并给出特定情形下分母的不可约性。
Arithmetic Constraints and Limit Laws for Diagonal Rational Partitions
本文研究对角有理分拆的计数渐近与极限分布，给出主项及修正项，并证明素数分母块总和的阈值行为及残差分布的高斯极限。
Approximating $1$ by Sums of Unit Fractions
研究正整数有限多重集\(A\)，埃尔德什和格雷厄姆曾给出相关结论并提问，本文证明了对于足够大的\(K\)，\(\varepsilon(A)\leq \exp\bigl(-c\sqrt{K\log K}\bigr)\)这一拉伸指数估计。
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