Discretized sum-product type problems: Energy variants and Applications
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866929292658606080 |
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| author | Pham, Quy Pham, Thang Shen, Chun-Yen |
| author_facet | Pham, Quy Pham, Thang Shen, Chun-Yen |
| contents | In this paper, we provide estimates for the additive discretized energy of \[\sum_{c\in C} |\{(a_1, a_2, b_1, b_2)\in A^2\times B^2: |(a_1 +cb_1) - (a_2 + cb_2)|\le δ\}|_δ,\] that depend on non-concentration conditions of the sets. Our proof follows the Guth-Katz-Zahl approach (2021) with appropriate changes along the way clarifying and optimizing many of the steps. Several applications will also be discussed. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2211_02277 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Discretized sum-product type problems: Energy variants and Applications Pham, Quy Pham, Thang Shen, Chun-Yen Classical Analysis and ODEs Combinatorics Number Theory In this paper, we provide estimates for the additive discretized energy of \[\sum_{c\in C} |\{(a_1, a_2, b_1, b_2)\in A^2\times B^2: |(a_1 +cb_1) - (a_2 + cb_2)|\le δ\}|_δ,\] that depend on non-concentration conditions of the sets. Our proof follows the Guth-Katz-Zahl approach (2021) with appropriate changes along the way clarifying and optimizing many of the steps. Several applications will also be discussed. |
| title | Discretized sum-product type problems: Energy variants and Applications |
| topic | Classical Analysis and ODEs Combinatorics Number Theory |
| url | https://arxiv.org/abs/2211.02277 |