Tom Sanders
We show that if A is a subset of Z/pZ (p a prime) of density bounded away from 0 and 1 then the A(Z/pZ)-norm (that is the l^1-norm of the Fourier transform) of the characterstic function of A is bounded below by an absolute constant times (log p)^{1/2 - ε} as p tends to infinity. This improves on the exponent 1/3 in recent work of Green and Konyagin.
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Let p be a prime, and let f : Z/pZ -> R be a function with average value 0 and ||f||_A <= 1, where ||f||_A denotes the algebra norm (L^1 norm of the Fourier transform). Then f(x) is small for some x, specifically min_x |f(x)| is no more than O(log p)^{-1/3 + eps}. One should think of f as being ``approximately continuous''; our result is then an ``approximate intermediate value theorem''. As an immediate consequence we show that if B in Z/pZ is a set of cardinality (p-1)/2 then the algebra norm ||1_B||_A is >> (log p)^{1/3 - eps}. This gives a result on a ``mod p'' analogue of Littlewood's wel…
We show that if $A$ is a finite set of non-negative integers then the number of zeros of the function \[ f_A(θ) = \sum_{a \in A} \cos(aθ), \] in $[0,2π]$, is at least $(\log \log \log |A|)^{1/2-\varepsilon}$. This gives the first unconditional lower bound on a problem of Littlewood, solves a conjecture of Borwein, Erdélyi, Ferguson and Lockhart and improves upon work of Borwein and Erdélyi. We also prove a result that applies to more general cosine polynomials with "few" distinct rational coefficients. One of the main ingredients in the proof is perhaps of independent interest: we show that if…
The Hardy-Littlewood majorant problem asks whether L^p norms of functions on the circle grow if one replaces their Fourier coefficients with their absolute values. This is clear if p is an even integer, but false if p is any other number. One can still ask if the norm grows at most by the degree raised to a small power for any p. We show that this is so with any epsilon power provided the majorizing function is the Dirichlet kernel on a random set, with large probability.
Littlewood raised the question of how slowly ||f_n||_4^4-||f_n||_2^4 (where ||.||_r denotes the L^r norm on the unit circle) can grow for a sequence of polynomials f_n with unimodular coefficients and increasing degree. The results of this paper are the following. For g_n(z)=\sum_{k=0}^{n-1}e^{πik^2/n} z^k the limit of (||g_n||_4^4-||g_n||_2^4)/||g_n||_2^3 is 2/π, which resolves a mystery due to Littlewood. This is however not the best answer to Littlewood's question: for the polynomials h_n(z)=\sum_{j=0}^{n-1}\sum_{k=0}^{n-1} e^{2πijk/n} z^{nj+k} the limit of (||h_n||_4^4-||h_n||_2^4)/||h_n||…
Consider a trigonometric polynomial f of degree N, and associate to it the polynomial F in which each coefficient of f is replaced by its absolute value. F is called the majorant of f. We show that the L^3 norm of f can be larger than that of F by a power of N. Similar examples could be constructed in which 3 is replaced by any p > 2 not equal to an even integer. This provides a negative answer to a question, the "Hardy-Littlewood majorant problem", which has been the object of some study. Had the answer been affirmative, there would have followed exciting results connected with the restrictio…
On an Erdös Problem about the Maximum Modulus of Littlewood Polynomials on the Unit Circl
该研究针对1957年的Erdös问题，证明系数仅取±1的n次Littlewood多项式，其在单位圆上的最大模平方存在下界，推进了相关问题的解决
We give $L^1$-norm estimates for exponential sums of a finite sets $A$ consisting of integers or lattice points. Under the assumption that $A$ possesses sufficient multidimensional structure, our estimates are stronger than those of McGehee-Pigno-Smith and Konyagin. These theorems improve upon past work of Petridis.
We use Jacob's ladders to solve the fine problem how to divide of the Hardy-Littlewood integral to equal parts, for example of magnitude $h=6.6\times 10^{-27}$ (the numerical value of elementary Planck quantum). The result of the paper cannot be obtained in known theories of Balasubramanian, Heath-Brown and Ivic.
An improved lower bound for a problem of Littlewood on the zeros of cosine polynomials
针对 Littlewood 关于余弦多项式最少零点数的问题，本文证明了新下界 $Z(N)\geqslant(\log\log N)^{(1+o(1))}$，相较 Erdélyi 和 Sahasrabudhe 的三重对数界实现指数级改进。
We introduce a wide class of deterministic subsets of primes of zero relative density and we prove Roth's Theorem in these sets, namely, we show that any subset of them with positive relative upper density contains infinitely many non-trivial three-term arithmetic progressions. We also prove that the Hardy--Littlewood majorant property holds for these subsets of primes. Notably, our considerations recover the results for the Piatetski--Shapiro primes for exponents close to $1$, which are primes of the form $\lfloor n^c\rfloor$ for a fixed $c>1$.
The Hardy-Littlewood majorant problem was raised in the 30's and it can be formulated as the question whether $\int |f|^p\ge \int|g|^p$ whenever $\hat{f}\ge|\hat g|$. It has a positive answer only for exponents $p$ which are even integers. Montgomery conjectured that even among the idempotent polynomials there must exist some counterexamples, i.e. there exists some finite set of exponentials and some $\pm$ signs with which the signed exponential sum has larger $p^{\rm th}$ norm than the idempotent obtained with all the signs chosen + in the exponential sum. That conjecture was proved recently …
The Littlewood problem and non-harmonic Fourier series
针对 Littlewood 问题，本文建立非整数频率三角多项式在大区间上的 $L^1$ 范数定量估计，推广并统一了多项经典结果。
The aim of this paper is to exhibit a wide class of sparse deterministic sets, $\mathbf B \subseteq \mathbb{N}$, so that \[ \limsup_{N \to \infty} N^{-1}|\mathbf B \cap [1,N]|= 0, \] for which the Hardy--Littlewood majorant property holds: \[ \sup_{|a_n|\le 1} \Big\| \sum_{n\in\mathbf B\cap[1, N]} a_n e^{2 πi n ξ}\Big \|_{L^p(\mathbb{T}, {\mathrm d} ξ)} \leq \mathbf{C}_p \Big\| \sum_{n\in\mathbf B\cap[1, N]} e^{2 πi n ξ} \Big\|_{L^p(\mathbb{T}, {\mathrm d} ξ)}, \] where $p \geq p_{\mathbf{B}}$ is sufficiently large, the implicit constant $\mathbf{C}_p$ is independent of $N$, and the supremum i…
We give the full solution of the following problem: obtain sharp inequalities between the moduli of smoothness $ω_α(f,t)_q$ and $ω_β(f,t)_p$ for $0<p<q\le \infty$. A similar problem for the generalized $K$-functionals and their realizations between the couples $(L_p, W_p^ψ)$ and $(L_q, W_q^φ)$ is also solved. The main tool is the new Hardy-Littlewood-Nikol'skii inequalities. More precisely, we obtained the asymptotic behavior of the quantity $$ \sup_{T_n} \frac{\Vert \mathcal{D}(ψ)(T_n)\Vert_q}{\Vert \mathcal{D}(φ)(T_n)\Vert_p},\qquad 0<p<q\le \infty, $$ where the supremum is taken over all no…
A Littlewood-Paley approach to the Mittag-Leffler function in the frequency space and applications to nonlocal problems
本文通过Littlewood-Paley理论研究频率空间中Mittag-Leffler函数的Lebesgue指数p(p(γ))，并推导了非局部空间-时间问题中的关键估计。
Distribution of mixed character sums and extremal problems for Littlewood polynomials
本文研究混合特征和的分布特性，并应用结果构造具有大Mahler测度和L1范数的Littlewood多项式，同时证明Turyn多项式L_{2k}范数在α=1/4处渐近最小，解决了Günther和Schmidt的猜想。
We study the problem concerning the variation of the Hardy-Littlewood maximal function in higher dimensions. As the main result, we prove that the variation of the non-centered Hardy-Littlewood maximal function of a radial function is comparable to the variation of the function itself.
The norm of the centered Hardy--Littlewood maximal operator
本文确定了中心 Hardy--Littlewood 极大算子在实数轴上的精确 $L^p$ 范数为 $p/(p-1)$，解决了尖锐强型问题，并否证了相关猜想。
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