An update on the sum-product problem

We improve the best known sum-product estimates over the reals. We prove that \[\max(|A+A|,|A+A|)\geq |A|^{\frac{4}{3} + \frac{2}{1167} - o(1)}\,,\] 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.

EVALUASI TARGET DAN RENCANA KERJA KEGIATAN KADER MASYARAKAT DI DESA CIAPUS, KECAMATAN CIOMAS, KABUPATEN BOGOR

PKM is a collaboration with Ambissi Association partners who have formed 23 cadres of Ciapus village. The activities are Health Services, Children's Reading Gardens, Savings and Loans, Medicinal Plant and Culinary Product. Problem analysis is related to the targets and work plans that have been made in the PKM of 2018-2019 period last semester. PKM problems now with the aim of seeing the realization of targets and work plans that have been made and to strengthen the understanding of cadres' self-evaluation of achievement thinking patterns. Due to the Covid-19 pandemic, activities were carried out by distributing a questionnaire "Am I ready to carry out target-oriented activities". The theoretical basis uses the need of achievement from McCleland combined with the concept of "Achievement Thinking". There are 19 cadres. The target evaluation shows that the Health, Culinary and Medicinal Plants group has succeeded in achieving the target. Evaluation of the work plan shows that although all cadres are actively working, the weakness of the group is increasing. The results of the assessment of the readiness to carry out target orientation activities were nine cadres of the prestigious type, eight cadres of the affiliative type and two cadres who did not describe their type profile. The internet network is very weak, interaction via the internet / zoom cannot be carried out. This activity uses Rp7,500,000 which are mostly used to increase development capital in five Cadre activities. Outcomes and achievements in the form of the article in the Journal and Copyright of Questionnaire.ABSTRAK: PKM merupakan kerjasama dengan mitra Perkumpulan Ambissi yang selama ini telah membentuk kader desa Ciapus sebanyak 23 orang. Kegiatan yang dilakukan meliputi Pelayanan Kesehatan, Taman Baca Anak, Kegiatan Simpan Pinjam, Kegiatan Tanaman Obat dan Kegiatan Produk Kuliner. Analisis permasalahan dikaitkan dengan target dan rencana kerja yang teah dibuat pada PKM periode 2018-2019 semester genap lalu. Permasalahan PKM sekarang dengan tujuan melihat realisasi target dan rencana kerja yang telah dibuat. Disamping itu untuk menguatkan pemahaman evaluasi diri kader mengenai pola berpikir prestatif. Pada awalnya evaluasi ini akan dilakukan dalam bentuk pelatihan karena kondisi pandemi Covid-19 kegiatan dilakukan dengan menyebarkan kuesioner “Siapkah saya menjalankan kegiatan berorientasi target”. Dasar teori menggunakan need of achievement dari McClelland yang digabungkan dengan konsep “Pola Berpikir Prestatif”. Jumlah kader aktif sebanyak 19 orang. Evaluasi target memperlihatkan kelompok Pelayanan Kesehatan, Kuliner dan Tanaman Obat telah berhasil mencapai target. Evaluasi rencana kerja memperlihatkan walaupun semua kader aktif bekerja tapi kelemahan kelompok bertambah. Hasil asesmen kesiapan menjalankan kegiatan orientasi target terdapat 9 kader bertipe prestatif, 8 kader bertipe afiliatif dan 2 kader kurang menggambarkan profil tipenya. Mengingat lokasi di desa jaringan internet sangat lemah m aka tidak dapat dilakukan interaksi melalui internet/zoom. Kegiatan menggunakan dana sebanyak Rp 7.500.000 yang sebagian besar digunakan untuk menambah modal pengembangan di 5 kegiatan Kader. Luaran dan capaian dalam bentuk artikel “Evaluasi Target dan Rencana Kerja Kegiatan Kader Masyarakat Desa Ciapus, kecamatan Ciomas, Kabupataen Bogor” di Jurnal dan Hak Cipta Kuesioner “Kesiapan Menjalankan Kegiatan Berorientasi Target untuk Kader Masyarakat Desa”

POLYNOMIAL VERSION ON THE SUM-PRODUCT PROBLEM

This work is about the generalization of sum-product problem. The general principle of it was formulated in the Erdos-Szemeredi’s hypothesis. Instead of the Minkowski sum in this hypothesis, the set of values f(x,y) of a homogeneous polynomial f lin two variables, where x and y belong to subgroup G of is considered. The lower bound on the cardinality of such set is obtained. This topic has an applied value in the theory of information and dynamics in calculating the probabilities of events, as well as in various methods of encoding and decoding information.

On the Discretized Sum-Product Problem

We give a new proof of the discretized ring theorem for sets of real numbers. As a special case, we show that if $A\subset \mathbb{R}$ is a $(\delta ,1/2)_1$-set in the sense of Katz and Tao, then either $A+A$ or $A.A$ must have measure at least $|A|^{1-\frac{1}{68}}$.

Numerical study of models of optimal distribution of recreational resources

Numerical realization of the problems of optimal distribution of recreational resources adapted to the continuous non-linear multiproduct problem of optimal partitioning of sets, the solution of which can be interesting in the sense of the development of recreation and tourism, is carried out. The success of most management tasks depends on the best way to use resources. In general, this problem is reduced to the problems of optimal resource allocation. The tasks of this type include the tasks of optimal placement of tourist complexes on a given territory, which is attractive in the sense of the tourism industry. In this case, the models of functioning and development of tourist and recreational systems are considered that can be used for analysis and forecasting of the tourism industry, both at the regional and national and international levels. The problem of the optimal distribution of recreational resources in the conditions of the modern and perspective structure of recreational needs was investigated. A mathematical model of the problem is constructed, which is a nonlinear continuous multi-product problem of the optimal partition of a set into its disjoint subsets (among which there may be empty ones) with fixed coordinates of the centers of these subsets under constraints in the form of equalities and inequalities. An analytical solution of the problem is obtained, which includes parameters that are sought as the optimal solution of the auxiliary dual finite-dimensional optimization problem with a nonsmooth target functional. On the basis of the analytical solution of the problem studied, a theoretically grounded solution algorithm was developed. A numerical investigation of the problem is carried out. The presented research results can serve as a useful tool in the sense of effective optimal partitioning of the functioning zones of existing tourist and recreational facilities that are able to produce service complexes to meet the necessary needs of recreants from the service area in conditions of the optimal cost of recreational needs

Expansion for cubes in the Heisenberg group

In the last decade the growth in groups focused the interest of many researchers in the domain of additive combinatorics. In this note, we investigate the special case of the Heisenberg group {\mathcal{H}} of order 3 on different fields R, namely, {R=\mathbf{R}} and {R=\mathbf{F}_{p}} . Denoting by {[x,y,z]} the elements of {\mathcal{H}} , a cube is a subset of the type {[A,B,C]} , with {A,B,C\subset R} . This study relies to the well-known sum-product problem and its extensions, from which we obtain strong lower bounds for the size of the square {[A,B,C]^{2}} of cubes in {\mathcal{H}} . In the final section, we also consider the problem of counting the subsets of {\mathcal{H}} of the type {[A,B,C]^{2}} .