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Vjekoslav Kovač

Publications and source records attributed to Vjekoslav Kovač.

At least 19 recordsLinked to original sources
arXiv2026

Estimates for $L^p$ variants of Gowers norms

Motivated by a log-convexity question of Bennett and Tao, we consider Gowers-type functionals defined by $L^p$ norms of multiple autocorrelations. We prove sharp bounds in terms of $L^q$ norms and characterise the near-extremisers on Euclidean spaces as well as locally compact abelian groups. We also establish two families of degree-lowering inequalities and study their near-extremisers. As a byproduct of this broader theory, we show that the constant in the log-convexity estimate for Gowers norms is strictly less than unity, confirming the aforementioned conjecture of Bennett and Tao. Finally, we characterise the $p$ for which the Gowers-type functionals necessarily satisfy the triangle inequality on nonnegative measurable functions.

math.CA↗
arXiv2026

Eventually greedy best Egyptian underapproximations of rational numbers via optimal control

We prove that every positive rational number has eventually greedy best Egyptian underapproximations, both when repetitions of the denominators are allowed and when the denominators are required to be distinct. This answers affirmatively a problem originating with Erdős and Graham and later revisited by Nathanson, and yields an application concerning the maximal asymptotic growth of denominators in unit fraction series converging to a given rational number. We reformulate the question as an optimal control problem for a dynamical system, construct an appropriate payoff function, and study properties of the associated Bellman function. We also answer another question of Nathanson by constructing an irrational number with unique and greedy best Egyptian underapproximations.

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arXiv2026

A converse to generalized Runcorn's theorem

We study an inverse problem for generalized Runcorn's theorem motivated by an apparent paradox of lunar magnetism. In mathematical terms, we characterize square-integrable complex functions $f$ such that $$ \int_{\{ x\in\mathbb{R}^n : a \leq |x| \leq b \}} f(x) \nabla u(x)\cdot\nabla v(x)\,\mathrm{d}\mathrm{V}(x) = 0 $$ for every complex harmonic function $u$ in the inner ball $|x| r_-$ that vanishes at infinity. The solution space depends on whether the radii $a$ and $b$ such that $r_-<a<b<r_+$ are regarded as varying or fixed. The solutions are described through the expansion of $f$ into spherical harmonics, and explicit representations are provided.

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arXiv2026

Irrationality of rapidly converging series: a problem of Erdős and Graham

Answering a question of Erdős and Graham, we show that the double exponential growth condition $\limsup_{n\to\infty}a_n^{1/ϕ^n}=\infty$ for a strictly increasing sequence of positive integers $\{a_n\}_{n=1}^\infty$ is sufficient for the series $\sum_{n=1}^\infty 1/(a_n a_{n+1})$ to have an irrational sum; here $ϕ$ denotes the golden ratio. We also provide a positive generalization to $\sum_{n=1}^\infty 1/(a_n^{w_0}\cdots a_{n+d-1}^{w_{d-1}})$, and a negative result showing that some of its instances are essentially optimal. The original problem was autonomously solved by the AI agent \emph{Aletheia}, powered by Gemini Deep Think, while the remaining material is largely a product of human-AI interactions.

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arXiv2026

Rectangles, triangles and Schrödinger waves

Can a finite set of lattice points determine many rectangles and few isosceles triangles? This turns out to be a surprisingly interesting question in combinatorial geometry that we answer using basic analytic number theory combined with a finite-field construction. The result is useful because it gives obstructions to Mizohata--Takeuchi-type estimates in the setting of the paraboloid. Specifically, we establish transference between Euclidean and periodic weighted $\mathrm{L}^2$ estimates for solutions to the Schrödinger equation, and then relate the failure of the latter to quantities tied to combinatorial problems, such as the one above. By completing this programme we give new explicit combinatorial counterexamples to the paraboloid case of the Mizohata--Takeuchi conjecture, which was recently shown to be false by Cairo for curved hypersurfaces.

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arXiv2026

On hyperbolic corners and unit-area triangles in planar sets of large measure

For large $R$, we consider measurable sets $A\subseteq [0,R]^2$ that avoid triples of points of the form $(x,y)$, $(x+t,y)$, $(x,y+1/t)$ with $x,y\in\mathbb{R}$ and $t>0$, i.e., the vertices of upward-oriented, axis-aligned right triangles of area $1/2$. We prove that the measures of such sets satisfy $|A|= O_c(R^2/(\log R)^c)$ for any constant $c<1/4$. An ingredient in the proof is a hyperbolic variant of the two-dimensional trilinear smoothing inequality by Christ, Durcik, and Roos. The aforementioned upper bound is complemented with an example of a set of measure $Ω(R\log R)$ avoiding the same point configuration. Next, we study measurable sets $A\subseteq [0,R]^2$ that avoid triples of points spanning a triangle of a given fixed area and establish a sharpening of the aforementioned upper bound to any $c<1/2$. This makes partial progress on a question by Erdős, who conjectured an upper bound $O(1)$, and improves over a quantitatively weak $o(R^2)$ result by Graham. The latter proof additionally uses induction on scales to interchangeably control the density and the Riesz energy of the set $A$.

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arXiv2026

A Banach space that distinguishes two maximal operators

Maz'ya and Shaposhnikova introduced a non-classical maximal operator $M^\diamond$ as the maximal convolution with the vector-valued signum kernel truncated to centered balls. We construct a translation-invariant Banach space of locally integrable functions on which $M^\diamond$ is bounded, but the sharp maximal operator $M^\sharp$ is not. This answers one of Maz'ya's questions from a collection of 75 open problems in analysis.

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arXiv2026

Near-optimal density theorems for large dilates of large point configurations

We study density thresholds that force a measurable set $E\subseteq\mathbb{R}^d$ to contain all sufficiently large similar copies of every $n$-point configuration. We prove a lower bound of the form $1-O((\log n)/n)$, which matches the known upper bound up to the logarithmic factor, thus essentially resolving a problem posed by Falconer, Yavicoli, and the first author of the present paper. We also study the same problem for embeddings of $n$-point configurations into $\mathbb{R}^d$ equipped with the $\ell^p$ norm, obtaining an asymptotically sharp bound $1-1/n+o(1/n)$, as soon as $p\in(1,\infty)\setminus\{2\}$. In the proof of the former estimate we use equidistribution of polynomial sequences modulo $1$ combined with probabilistic thinning. The proof of the latter estimate relies on the geometry of the $\ell^p$ spaces for $p\neq2$.

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arXiv2026

Sharp variational inequalities for the Hardy-Littlewood maximal operator on finite undirected graphs

We study sharp $p$-variational inequalities for the Hardy-Littlewood maximal operator on complete graphs, answering in the affirmative a question by Feng Liu and Qingying Xue. We also use computational assistance to find sharp constants in $1$-variational inequalities for all connected graphs on at most five vertices and pose a conjecture on the corresponding sharp constants for path graphs. Finally, we construct finite graphs with arbitrarily large $p$-variational constants.

math.CA↗
arXiv2026

On the number of divisors of Mersenne numbers

Denote $f(n):=\sum_{1\le k\le n} τ(2^k-1)$, where $τ$ is the number of divisors function. Motivated by a question of Paul Erdős, we show that the sequence of ratios $f(2n)/f(n)$ is unbounded. We also present conditional results on the divergence of this sequence to infinity. Finally, we test numerically both the conjecture $f(2n)/f(n)\to\infty$ and our sufficient conditions for it to hold.

math.NT↗
arXiv2026

Coloring and density theorems for configurations of a given volume

This is a treatise on finite point configurations spanning a fixed volume to be found in a single color-class of an arbitrary finite (measurable) coloring of the Euclidean space $\mathbb{R}^n$, or in a single large measurable subset $A\subseteq\mathbb{R}^n$. More specifically, we study vertex-sets of simplices, rectangular boxes, and parallelotopes, attempting to make progress on several open problems posed in the 1970s and the 1980s. As one of the highlights, we give a negative answer to a question of Erdős and Graham, by coloring the Euclidean plane $\mathbb{R}^2$ in $25$ colors without creating monochromatic rectangles of unit area. More generally, we construct a finite coloring of the Euclidean space $\mathbb{R}^n$ such that no color-class contains the $2^m$ vertices of any (possibly rotated) $m$-dimensional rectangular box of volume $1$. A positive result is still possible if rectangular boxes of merely sufficiently large volumes are sought in a single color-class of a finite measurable coloring of $\mathbb{R}^n$, and we establish it under an additional assumption $n\geq m+1$. Also, motivated by a question of Graham on reasonable bounds in his result on monochromatic axes-aligned right-angled $m$-dimensional simplices, we establish its measurable coloring and density variants with polylogarithmic bounds, again in dimensions $n\geq m+1$. Next, we generalize a result of Erdős and Mauldin, by constructing an infinite measure set $A\subseteq\mathbb{R}^n$ such that every $n$-parallelotope with vertices in $A$ has volume strictly smaller than $1$. Finally, some results complementing the literature on isometric embeddings of hypercube graphs and on the hyperbolic analogue of the Hadwiger-Nelson problem also follow as byproducts of our approaches.

math.CO↗
arXiv2025

Lacunary sequences whose reciprocal sums represent all rational numbers in an interval

Disproving a conjecture of Bleicher and Erdős, we show that there exists a lacunary sequence of positive integers such that finite sums of reciprocals of its terms attain all rational numbers from a non-empty open interval. We also study several stronger variants of their original problem: determining the value of the optimal lacunarity parameter, representing rational numbers infinitely many times, finding such lacunary sequences with arbitrarily large jumps, and relating the maximal length of a filled interval to a prescribed lacunarity parameter.

math.NT↗
arXiv2025

Polygons of unit area with vertices in sets of infinite planar measure

Paul Erdős and R. Daniel Mauldin asked a series of questions on certain types of polygons of area $1$, the vertices of which can be found in every planar set of infinite Lebesgue measure. We address two of these questions, one on cyclic quadrilaterals and the other on convex polygons with congruent sides, with respectively positive and negative answers.

math.CA↗
arXiv2025

On several irrationality problems for Ahmes series

Using basic tools of mathematical analysis and elementary probability theory we address several problems on the irrationality of series of distinct unit fractions, $\sum_k 1/a_k$. In particular, we study subseries of the Lambert series $\sum_k 1/(t^k-1)$ and two types of irrationality sequences $(a_k)$ introduced by Paul Erdős and Ronald Graham. Next, we address a question of Erdős, who asked how rapidly a sequence of positive integers $(a_k)$ can grow if both series $\sum_k 1/a_k$ and $\sum_k 1/(a_k+1)$ have rational sums. Our construction of double exponentially growing sequences $(a_k)$ with this property generalizes to any number $d$ of series $\sum_k 1/(a_k+j)$, $j=0,1,2,\ldots,d-1$, and, in particular, also gives a positive answer to a question of Erdős and Ernst Straus on the interior of the set of $d$-tuples of their sums. Finally, we prove the existence of a sequence $(a_k)$ such that all well-defined sums $\sum_k 1/(a_k+t)$, $t\in\mathbb{Z}$, are rational numbers, giving a negative answer to a conjecture by Kenneth Stolarsky.

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arXiv2025

Inequalities in Fourier analysis on binary cubes

This paper studies two classical inequalities, namely the Hausdorff-Young inequality and equal-exponent Young's convolution inequality, for discrete functions supported in the binary cube $\{0,1\}^d\subset\mathbb{Z}^d$. We characterize the exact ranges of Lebesgue exponents in which sharp versions of these two inequalities hold, and present several immediate consequences. First, if the functions are specialized to be the indicator of some set $A\subseteq\{0,1\}^d$, then we obtain sharp upper bounds on two types of generalized additive energies of $A$, extending the works of Kane-Tao, de Dios Pont-Greenfeld-Ivanisvili-Madrid, and one of the present authors. Second, we obtain a sharp binary variant of the Beckner-Hirschman entropic uncertainty principle, as well as a sharp lower estimate on the entropy of a sum of two independent random variables with values in $\{0,1\}^d$. Finally, the sharp binary Hausdorff-Young inequality also reveals the exact range of dimension-free estimates for the Fourier restriction to the binary cube.

math.CA↗
arXiv2025

On the irrationality of certain super-polynomially decaying series

We give a negative answer to a question by Paul Erdős and Ronald Graham on whether the series \[ \sum_{n=1}^{\infty} \frac{1}{(n+1)(n+2)\cdots(n+f(n))} \] has an irrational sum whenever $(f(n))_{n=1}^{\infty}$ is a sequence of positive integers converging to infinity. To achieve this, we generalize a classical observation of Sōichi Kakeya on the set of all subsums of a convergent positive series. We also discuss why the same problem is likely difficult when $(f(n))_{n=1}^{\infty}$ is additionally assumed to be increasing.

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arXiv2024

Sharp estimates for Gowers norms on discrete cubes

We study optimal dimensionless inequalities $$ \|f\|_{U^k} \leq \|f\|_{\ell^{p_{k,n}}} $$ that hold for all functions $f\colon\mathbb{Z}^d\to\mathbb{C}$ supported in $\{0,1,\ldots,n-1\}^d$ and estimates $$ \|1_A\|_{U^k}^{2^k}\leq |A|^{t_{k,n}} $$ that hold for all subsets $A$ of the same discrete cubes. A general theory, analogous to the work of de Dios Pont, Greenfeld, Ivanisvili, and Madrid, is developed to show that the critical exponents are related by $p_{k,n} t_{k,n} = 2^k$. This is used to prove the three main results of the paper: an explicit formula for $t_{k,2}$, which generalizes a theorem by Kane and Tao, two-sided asymptotic estimates for $t_{k,n}$ as $n\to\infty$ for a fixed $k\geq2$, which generalize a theorem by Shao, and a precise asymptotic formula for $t_{k,n}$ as $k\to\infty$ for a fixed $n\geq2$.

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