arXiv subjects

Misha Rudnev

Publications and source records attributed to Misha Rudnev.

At least 19 recordsLinked to original sources

arXiv2026

Multiplicative Subgroups of Prime Fields Are Not Sumsets

Let $H \leq \mathbb{F}_p^*$ be a proper multiplicative subgroup, and suppose that $H = A+B$ for some $A,B \subseteq \mathbb{F}_p$. We prove that either one of the summands is a singleton, or $|A|=|B|=2$ and $|H|=4$. In particular, no proper multiplicative subgroup of $\mathbb{F}_p^*$ can be written as $A+B$ with $|A|,|B|>2$. Our proof builds on the Hanson-Petridis polynomial method and Kalmynin's subsequent resolution of S\'ark\"ozy's conjecture for quadratic residues. Using Kalmynin's $|A|=|B|$ theorem as a structural input, we develop uniform combinatorial and arithmetic arguments which apply to multiplicative subgroups of arbitrary index.

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arXiv2024

On Distinct Angles in the Plane

We prove that if $N$ points lie in convex position in the plane then they determine $\Omega(N^{5/4})$ distinct angles, provided that the points do not lie on a common circle. This is derived from a more general claim that if $N$ points in the convex position in the real plane determine $KN$ distinct angles, then $K=\Omega(N^{1/4})$ or $\Omega(N/K)$ points are co-circular. The proof makes use of the implicit order one can give to points in convex position and relies on a slightly more general order assumption. The assumption enables one to reduce the issue to counting incidences between points and a multiset of cubic curves, with special attention being paid to the case when the curves are reducible.

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arXiv2023

The sum-product problem for integers with few prime factors

It was asked by E. Szemer\'edi if, for a finite set $A\subset\mathbb{Z}$, one can improve estimates for $\max\{|A+A|,|A\cdot A|\}$, under the constraint that all integers involved have a bounded number of prime factors -- that is, each $a\in A$ satisfies $\omega(a)\leq k$. In this paper, answer Szemer\'edi's question in the affirmative by showing that this maximum is of order $|A|^{\frac{5}{3}-o(1)}$ provided $k\leq (\log|A|)^{1-\epsilon}$ for some $\epsilon>0$. In fact, this will follow from an estimate for additive energy which is best possible up to factors of size $|A|^{o(1)}$.

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arXiv2021

An update on the sum-product problem

We improve the best known sum-product estimates over the reals. We prove that \[ \max(|A+A|,|AA|)\geq |A|^{\frac{4}{3} + \frac{2}{1167} - o(1)}\,, \] for a finite $A\subset \mathbb R$, following a streamlining of the arguments of Solymosi, Konyagin and Shkredov. We include several new observations to our techniques. Furthermore, \[ |AA+AA|\geq |A|^{\frac{127}{80} - o(1)}\,. \] Besides, for a convex set $A$ we show that \[ |A+A|\geq |A|^{\frac{30}{19}-o(1)}\,. \] This paper is largely self-contained.

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arXiv2021

On the Pinned Distances Problem in Positive Characteristic

We study the Erd\H os-Falconer distance problem for a set $A\subset \mathbb{F}^2$, where $\mathbb{F}$ is a field of positive characteristic $p$. If $\mathbb{F}=\mathbb{F}_p$ and the cardinality $|A|$ exceeds $p^{5/4}$, we prove that $A$ determines an asymptotically full proportion of the feasible $p$ distances. For small sets $A$, namely when $|A|\leq p^{4/3}$ over any $\mathbb{F}$, we prove that either $A$ determines $\gg|A|^{2/3}$. For both large and small sets, the results proved are in fact for pinned distances.

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arXiv2021

On incidences of lines in regular complexes

A regular linear line complex is a three-parameter set of lines in space, whose Plücker vectors lie in a hyperplane, which is not tangent to the Klein quadric. Our main result is a bound $O(n^{1/2}m^{3/4} + m+n)$ for the number of incidences between $n$ lines in a complex and $m$ points in $\mathbb F^3$, where $\mathbb F$ is a field, and $n\leq char(\mathbb F)^{4/3}$ in positive characteristic. Zahl has recently observed that bichromatic pairwise incidences of lines coming from two distinct line complexes account for the nonzero single distance problem for a set of $n$ points in $\mathbb F^3$. This implied the new bound $O(n^{3/2})$ for the number of realisations of the distance, which is a square, for $\mathbb F$, where $-1$ is not a square in the $\mathbb F$-analogue of the Erd\H os single distance problem in $\mathbb R^3$. Our incidence bound yields, under a natural constraint, a weaker bound $O(n^{1.6})$, which holds for any distance, including zero, over any $\mathbb F$.

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arXiv2021

Higher Convexity and Iterated Second Moment Estimates

We prove bounds for the number of solutions to $$a_1 + \dots + a_k = a_1' + \dots + a_k'$$ over $N$-element sets of reals, which are sufficiently convex or near-convex. A near-convex set will be the image of a set with small additive doubling under a convex function with sufficiently many strictly monotone derivatives. We show, roughly, that every time the number of terms in the equation is doubled, an additional saving of $1$ in the exponent of the trivial bound $N^{2k-1}$ is made, starting from the trivial case $k=1$. In the context of near-convex sets we also provide explicit dependencies on the additive doubling parameters. Higher convexity is necessary for such bounds to hold, as evinced by sets of perfect powers of consecutive integers. We exploit these stronger assumptions using an idea of Garaev, rather than the ubiquitous Szemerédi-Trotter theorem, which has not been adapted in earlier results to embrace higher convexity. As an application we prove small improvements for the best known bounds for sumsets of convex sets under additional convexity assumptions.

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arXiv2021

Incidence bounds with Möbius hyperbolae in positive characteristic

We prove new incidence bounds between a plane point set, which is a Cartesian product, and a set of translates $H$ of the hyperbola $xy=λ\neq 0$, over a field of asymptotically large positive characteristic $p$. They improve recent bounds by Shkredov, which are based on using explicit incidence estimates in the early terminated procedure of repeated applications of the Cauchy-Schwarz inequality, underlying many qualitative results related to growth and expansion in groups. The improvement -- both quantitative, plus we are able to deal with a general $H$, rather than a Cartesian product -- is mostly due to a non-trivial "intermediate" bound on the number of $k$-rich Möbius hyperbolae in positive characteristic. In addition, we make an observation that a certain energy-type quantity in the context of $H$ can be bounded via the $L^2$-moment of the Minkowski distance in $H$ and can therefore fetch the corresponding estimates apropos of the Erdős distinct distance problem.

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arXiv2020

An Energy Bound in the Affine Group

We prove a nontrivial energy bound for a finite set of affine transformations over a general field and discuss a number of implications. These include new bounds on growth in the affine group, a quantitative version of a theorem by Elekes about rich lines in grids. We also give a positive answer to a question of Yufei Zhao that for a plane point set P for which no line contains a positive proportion of points from P, there may be at most one line, meeting the set of lines defined by P in at most a constant multiple of |P| points.

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arXiv2020

Higher convexity and iterated sum sets

Let $f$ be a smooth real function with strictly monotone first $k$ derivatives. We show that for a finite set $A$, with $|A+A|\leq K|A|$, $|2^kf(A)-(2^k-1)f(A)|\gg_k |A|^{k+1-o(1)}/K^{O_k(1)}$. We deduce several new sum-product type implications, e.g. that $A+A$ being small implies unbounded growth for a many enough times iterated product set $A \cdots A$.

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arXiv2020

On the number of hinges defined by a point set in $\mathbb R^2$

It is shown that the number of distinct types of three-point hinges, defined by a real plane set of $n$ points is $\gg n^2\log^{-3} n$, where a hinge is identified by fixing two pair-wise distances in a point triple. This is achieved via strengthening (modulo a $\log n$ factor) of the Guth-Katz estimate for the number of pair-wise intersections of lines in $\mathbb R^3$, arising in the context of the plane Erd\H os distinct distance problem, to a second moment incidence estimate. This relies, in particular, on the generalisation of the Guth-Katz incidence bound by Solomon and Sharir.

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arXiv2019

Bisector energy and pinned distances in positive characteristic

We prove a new lower bound for the number of pinned distances over finite fields: if $A$ is a sufficiently small subset of $\mathbb{F}_q^2$, then there is an element in $A$ that determines $\gg |A|^{2/3}$ distinct distances to other elements of $A$. Combined with results for large subsets $A\subseteq\mathbb{F}_q^2$, this improves all previously known lower bounds on distinct distances over finite fields. In fact, we obtain an upper bound for the number of isosceles triangles determined by $A$. For that we use the concept of bisector energy. It turns out that the latter can be expressed as a point-plane incidence bound, so one can use a theorem of the third author. The conversion to this incidence problem relies on the Blaschke-Grünwald kinematic mapping -- an embedding of the group of rigid motions of $\mathbb{F}_q^2$ into an open subset of the projective three space. This has long been known in kinematics and geometric algebra; we provide a proof for arbitrary fields using Clifford algebras.

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arXiv2019

New results on sum-product type growth over fields

We prove a range of new sum-product type growth estimates over a general field $\mathbb{F}$, in particular the special case $\mathbb{F}=\mathbb{F}_p$. They are unified by the theme of "breaking the $3/2$ threshold", epitomising the previous state of the art. These estimates stem from specially suited applications of incidence bounds over $\mathbb{F}$, which apply to higher moments of representation functions. We establish the estimate $|R[A]| \gtrsim |A|^{8/5}$ for cardinality of the set $R[A]$ of distinct cross-ratios defined by triples of elements of a (sufficiently small if $\mathbb{F}$ has positive characteristic, similarly for the rest of the estimates) set $A\subset \mathbb{F}$, pinned at infinity. The cross-ratio naturally arises in various sum-product type questions of projective nature and is the unifying concept underlying most of our results. It enables one to take advantage of its symmetry properties as an onset of growth of, for instance, products of difference sets. The geometric nature of the cross-ratio enables us to break the version of the above threshold for the minimum number of distinct triangle areas $Ouu'$, defined by points $u,u'$ of a non-collinear point set $P\subset \mathbb{F}^2$. Another instance of breaking the threshold is showing that if $A$ is sufficiently small and has additive doubling constant $M$, then $|AA|\gtrsim M^{-2}|A|^{14/9}$. This result has a second moment version, which allows for new upper bounds for the number of collinear point triples in the set $A\times A\subset \mathbb{F}^2$, the quantity often arising in applications of geometric incidence estimates.

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arXiv2019

On growth rate in $SL_2(\mathbf{F}_p)$, the affine group and sum-product type implications

This paper aims to study in more depth the relation between growth in matrix groups ${\rm SL_2}(\mathbf{F})$ and ${\rm Aff}(\mathbf{F})$ over a field $\mathbf{F}$ by multiplication and geometric incidence estimates, associated with the sum-product phenomenon over $\mathbf{F}$. It presents streamlined proofs of Helfgott's theorems on growth in the $\mathbf{F}_p$-case, which avoid sum-product estimates. For ${\rm SL_2}(\mathbf{F}_p)$, for sets exceeding in size some absolute constant, we improve the lower bound $\frac{1}{1512}$ for the growth exponent, due to Kowalski, to $\frac{1}{21}.$ For the affine group we fetch a sharp theorem of Szőnyi on the number of directions, determined by a point set in $\mathbf{F}_p^2$. We then focus on ${\rm Aff}(\mathbf{F})$ and present a new incidence bound between a set of points and a set of lines in $\mathbf{F}^2$, which explicitly depends on the energy of the set of lines as affine transformations under composition. This bound, strong when the number of lines is considerably smaller than the number of points, yields generalizations of structural theorems of Elekes and Murphy on rich lines in grids. In the special case when the set of lines is also a grid -- relating back to sum-products -- we use growth in ${\rm Aff}(\mathbf{R})$ to obtain a subthreshold estimate on the energy of the set of lines. This yields a unified way to break the ice in various threshold sum-product type energy inequalities. We show this in applications to energy estimates, corresponding to sets $A(A+ A)$, $A+AA$ (also embracing asymmetric versions) as well as $A+B$ when $A$ has small multiplicative doubling and $\sqrt{|A|} \le |B|\le|A|^{1+o(1)}$.

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arXiv2018

On the restriction problem for discrete paraboloid in lower dimension

We apply geometric incidence estimates in positive characteristic to prove the optimal $L^2 \to L^3$ Fourier extension estimate for the paraboloid in the four-dimensional vector space over a prime residue field. In three dimensions, when $-1$ is not a square, we prove an $L^2 \to L^{\frac{32}{9} }$ extension estimate, improving the previously known exponent $\frac{68}{19}.$

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arXiv2018

Stronger sum-product inequalities for small sets

Let $F$ be a field and a finite $A\subset F$ be sufficiently small in terms of the characteristic $p$ of $F$ if $p>0$. We strengthen the "threshold" sum-product inequality $$|AA|^3 |A\pm A|^2 \gg |A|^6\,,\;\;\;\;\mbox{hence} \;\; \;\;|AA|+|A+A|\gg |A|^{1+\frac{1}{5}},$$ due to Roche-Newton, Rudnev and Shkredov, to $$|AA|^5 |A\pm A|^4 \gg |A|^{11-o(1)}\,,\;\;\;\;\mbox{hence} \;\; \;\;|AA|+|A\pm A|\gg |A|^{1+\frac{2}{9}-o(1)},$$ as well as $$ |AA|^{36}|A-A|^{24} \gg |A|^{73-o(1)}. $$ The latter inequality is "threshold-breaking", for it shows for $ε>0$, one has $$|AA| \le |A|^{1+ε}\;\;\;\Rightarrow\;\;\; |A-A|\gg |A|^{\frac{3}{2}+c(ε)},$$ with $c(ε)>0$ if $ε$ is sufficiently small. This implies that regardless of $ε$, $$|AA-AA|\gg |A|^{\frac{3}{2}+\frac{1}{56}-o(1)}\,.$$

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arXiv2018

Point-plane incidences and some applications in positive characteristic

The point-plane incidence theorem states that the number of incidences between $n$ points and $m\geq n$ planes in the projective three-space over a field $F$, is $$O\left(m\sqrt{n}+ m k\right),$$ where $k$ is the maximum number of collinear points, with the extra condition $n< p^2$ if $F$ has characteristic $p>0$. This theorem also underlies a state-of-the-art Szemerédi-Trotter type bound for point-line incidences in $F^2$, due to Stevens and de Zeeuw. This review focuses on some recent, as well as new, applications of these bounds that lead to progress in several open geometric questions in $F^d$, for $d=2,3,4$. These are the problem of the minimum number of distinct nonzero values of a non-degenerate bilinear form on a point set in $d=2$, the analogue of the Erd\H os distinct distance problem in $d=2,3$ and additive energy estimates for sets, supported on a paraboloid and sphere in $d=3,4$. It avoids discussing sum-product type problems (corresponding to the special case of incidences with Cartesian products), which have lately received more attention.

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arXiv2017

On the few products, many sums problem

We prove new results on additive properties of finite sets $A$ with small multiplicative doubling $|AA|\leq M|A|$ in the category of real/complex sets as well as multiplicative subgroups in the prime residue field. The improvements are based on new combinatorial lemmata, which may be of independent interest. Our main results are the inequality $$ |A-A|^3|AA|^5 \gtrsim |A|^{10}, $$ over the reals, "redistributing" the exponents in the textbook Elekes sum-product inequality and the new best known additive energy bound $\mathsf E(A)\lesssim_M |A|^{49/20}$, which aligns, in a sense to be discussed, with the best known sum set bound $|A+A|\gtrsim_M |A|^{8/5}$. These bounds, with $M=1$, also apply to multiplicative subgroups of $\mathbb F^\times_p$, whose order is $O(\sqrt{p})$. We adapt the above energy bound to larger subgroups and obtain new bounds on gaps between elements in cosets of subgroups of order $Ω(\sqrt{p})$.

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